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THE PTIBLIC SCHOOL LATIN COmiSE.
j SCHOOL OF EDUCATION
LIBRARY
TEXTBOOK COLLECTION
GIFT OF
THE PUBLISHERS
' TiroUvi. ChownSTo. llhr.Sd.
<W« e&imot cloig this brief
notln wlaoDt th.TiHng Profewir Sxs-
m>T tor Msglng ttalH admlnble edition
« ft mat « nnlTenmUy read, vhlab he
hu enilelMd wlCb th« occiunulBted
■Man of Oa best moilem BOliolftrabip
. It I
Btnctont. It la, beyood
tha moM Dcmpleto, comn
Larly edltloo of Vibqil
rto appeu-ed Id thin ooiu
Kddihtionil Ti
London: LONGMANS, GREEN, & CO.
itrj."
. ^ V T > ^ '■> '■ ' \'- 7 M T O F ^
A
n:. - "-.IVCD
FLB2 4iti04.
v'
•l
> r^::r ANO STANFORD <
'.' -rTisnO-R UNIVERSITY.
I
V
i
r
AEITHMETIC
LIST OF EDUCATIONAL WOEKS
By the Et. Eby. JOHN W. COLENSO, late Bishop of Natal,
AEITHHETIO designed for the Use of SCHOOLS : to which Is added
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BY THB
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THE UTE MASTER OF TRINITY.
Extract from Dr. Whewell's Work on * A Liberal
Education/ pp. 168, 159.
As the basis of all real progress in Mathematics, boys ought
to acquire a good knowledge of Arithmetic and a habit of per-
forming the common operations of Arithmetic, and of applying
the rules in a correct and intelligent manner. This acquirement
appears to be often neglected at our most eminent classical
schools. Such a neglect is much to be regretted ; for the want
of this acquirement is a great practical misfortune, and is often
severely felt in after-life. Many persons who are supposed to
have received the best education which the country afibrds, are,
in all matters of numerical calculation, ignorant and helpless, in
a manner which places them, in this respect, far below the
members of the middle class, educated as they usually are.
Arithmetic is a matter of habit, and can be learnt only by long-
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)v v)fPAirrMENT OF ^
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PBEFACB-
Since this book was first published, some considerable
additions have been made to it, besides further modifica-
tions, with a view to correcting any defects which expe-
rience has from time to time detected, and bringing it up
to the requirements of the present day. These have been
carried out under my sanction and superintendence,
and to my entire satisfaction, by the Eev. J. Hunter,
formerly of the National Society's Training College,
Battersea, and chiefly at his suggestion ; and I consider
that the book has been much improved by them.
I have taken the opportunity, however, of my
being in England for a few weeks, to insert some addi-
tional pages on the Metric System of Weights and
Measures, the principles of which, by a rule of the
Council of Education in force in 1872, were required
to be taught to all children of Standards V. and VI.
in schools under the control of the Government. The
rule in question has, however, been since rescinded, as
requiring too much from elementary schools, while the
use of the Metric System has not yet been rendered
compulsory by Act of Parliament. But the general
adoption of that System in England is only, it seems
plain, a question of time.
J. W. NATAL.
London: December 2i, 1874.
TABLE OF CONTENTS.
I)
TRODTJCTION. — FlBST FOUB SiMPLE EuLES.
Definitions, Notation, and Numeration
Simple Addition ...
Subtraction -
Multiplication
Division
Answers to Examples
EUTHHETICAI. TaBLES
>xpouND Abithmetic aui the IIiqkeb Rules.
LJHAP.
I. Reduction ...
Compound Addition
Subtraction
Multiplication -
Division
Square and Cubic Measure
Miscellaneous Examples
II. Greatest Common Measure
Least Common Multiple -
III. Vulgar Fractions -
MiscELLAmous Examples
IV. Decimal Fractions -
Miscellaneous Examfies
V. Practice . - -
Miscellanjbous Examples
w
}i
it
Page
1
2
4
5
7
9
[9], [10]
- 11
- 14
. 17
- 19
- 21
. 25
■ 29
- 34
- 35
■ 38
■ Gi
' 67
- 71
« 1^
via TABLE OP CONTENTS.
Chap. Tago
VI. Proportion - - • * • - -81
Single Rule of Three - - - - - 84
Double Rule of Three - - - - - 94
VII. Interest - - - - - - -99
Discount ------- 105
Insurance, &c. - - - - - -107
Stocks - - - - - - - 109
Profit and Loss - - - - - - 112
Proportional Parts - - - - - -114
Chain Rule - - - - - - -118
Square Root and Cube Root - - - - 120
MiSCEIXANEOUS EXAMPLES - - - - - 126
Appendix I. Remarks on the Tables - - - -141
Appendix II. Decimal Coinage - - - - 149
Appendix III. The Metric System - - - - 156
Notes and Exahination-Papebs on Abithmetic.
Notes -----.- 167
Examination-Pa pers - - - - -178
Answers to the Examples ...... 209
Cambridge Examination-Papcr worked out - - - 231
AEITHMETia
Arithuetio is the science which treats of numbers — of the
mode of expressing them — of the manner of computing by
them — and of the various uses to which they are applied in
the practical business of life.
ITtie number one is called unity ; and an integer^ or whole
number, is a collection of ones, unities^ or units.
The figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, denote, respectively,
the numbers one, two, three, four, five, six, seven, eight, nine;
the figure 0, called zero or a cypher, expresses nought or
nothing ; but by means of these figures, which are called
the ten digits, or more commonly the nine digits and zero,
any number whatever can be expressed. This is effected
thus:
A figure standing bg itself, or on the right hand of other
figures, has its own proper value, expressing so many units;
A figure standing in the second place from the right is
considered to express so many tens of units ;
In the third place, so many tens of tens, or hundreds of
units ;
In the fourth place, so many tens of hundreds, or thou-
sands of units, &c., according to the following Table, called
the
NUMERATION TABLE.
712814367123419
g||goggggg|gg«
g ^ I i i i ^1
H P 2 o S
g
and 80 on to trillions^ quadrillion^^ &c. \{ rkeioeeeax^ .
2 SIMPLE ADDITION.
Notation is the art of expressing any given number hy
these figures ; Numeration the art of reading them, when so
expressed.
N.B. Examples in Notation and Numeration may be ob-
tained from those given in Addition and Subtraction.
The Romans used I for 1, V for 5, X for 10, L for 50, C for 100,
D or Iq for 500, M or CIo for 1000.
When any character was followed by one of less or equal value, the
expression denoted the sum of their simple valaes; but when preceded
by one of less ralue, the difference; thus III stood for 3, IV for 4, and
VI for 6, XL for 40, and LXX for 70, &c.
Every q annexed to Iq, and every C and 3 joined to CI^, increased
its value tenfold ; thus 1^3 stood for 5000, CCIqo for 10,000, &c.
A line drawn over a character increased its value a thousand-fold ;
thus V stood for 5000, C for 100,000.
The following sigiis are also made use of in Arithmetic :
+ (plus) shows that the number before which it stands is
to be added;
— {minus) that the number Ipefpre which it stands is to
be subtracted;
X (into) that the numbers between which it stands are to
be multiplied ;
-7- {bi/) that the number which stands before it is to be
divided by the one which follows ; and
= (equal) that the numbers between which it stands are
equal to each other.
Addition. — When any numbers are taken together, or
added, the resulting number is called their sum.
Ex. Add 94163 ^^ order to add whole numbers together, we
21954 ^^'st place them under one another, with their
7812 units-figures in the same vertical line 5 we then
593 add these figures thus, 6 and 7 are 12, and 3 are
^4895 ^^' ''"^ ^ "'^^ ^^* "''^ ^ ^''^ ^^' ^'^^ ^ ^''^ ^^' ^' ^'
24 units, or 2 tens and 4 units; we set the 4 under
1 65064 ^|j^ units-figures, to be the units-figure of the result,
Bod cany the 2 tens to be added to the second or tens column *, adding
I
SIMPLE ADDlllON. 3
this in the same manner, beginning with the 2 cariicd, thus 2 and 9 are
1 1, and 4 are 15, &c., we find the sum of the column to be 36, i. e. 36
ienSf or 3 tens of tens (i. c. 3 hundreds) and 6 tens; we set the 6 under
the tens-figures, to be the tens-figure of the result, and carry the 3
hundreds to the third or hundreds column : pursuing the same course lyith
this, wc find the sum of this column to be 40, L e. 40 hundreds or 4 tens
of hundreds (i. c. 4 thousands) and hundreds; we set the under the
hundreds-figures, to be the hundreds- figure of the result, and carry the 4
thousands, &c.
K.B. Any sums may be set at pleasure in Addition, and the
Answers proved by repeating the operation, beginning with the top
figure of the units column, when the result will be the same, if the sum
be worked correctly.
EXAMPLES IN ADDITION.
1. 321413
2. 543123
3. 536123
4. 123456
452734
234512
453215
234561
130421
713145
1234
345C12
3718
104234
4231
456123
24561
36142
51234
561234
341323
3451
613254
612345
6. 761284
6. 657890
7. 692387
8. 768453
612874
278679
4956
358428
8719
5798
87958
87?^
46759
67843
769378
54937
587999
489567
5790
495
987678
37429
87658
876578
9. -JiM^sf^ther five hundred and ninety-seven thousand six hundred
And eighty -five, forty-nine thousand three hundred and seven, four hun-
dred and ninOi thousand and sixty-seven, fourteen thousand and nineteen,
seven hundred thousand and seventy-four, sixty-five thousand and nine.
10. Add tdf^er seven hundred and seven thousand four hundred and
fifty. nine, ninfeifcy-cight thousand and seventy-four, six thousand eight
hundred and seven, five hundred thousand three hundred and nine,
seven thousand nine hundred and seventy-eight, nine hundred and nine
. thousand nine hundred and ninety-nine.
11. Add to^cMer fifty-five millions seven hundred thousand and five,
seven hundred millions nine hundred and eight thousand two hundred
and five, seventy-six millions fourteen thousand and fifty -nine, eight
hundred and seventy-seven millions nine hundred and two thousand and
forty-seven, seven millions eight hundred and four thousand five bsMvAxR.^
and twelve, five hundred and seventy -five ixnUioi\a e>\^\i Vxol^^ «5A.
CD6 thousand and' xunety-nine.
4 SIMPLE SUBTEACTlOJJ.
12. Add togetJier three hundred and nice millions four hundred and
seventeen thousand and eighty- seven, six hundred and seventj-five
thousand and forty-nine, seven thousand and ninety-seven millions
eight hundred and fourteen thousand three hundred and five, seventy*
nine millions five hundred and four thousand and forty-nine, six thousand
and seventy-eight millions four hundred and thirty-nine thousand six
hundred and forty-seven, seven thousand millions eight hundred and
seventy-six thousand four hundred and twenty-nine.
Subtraction. — When one number is taken from another,
or subtracted, the result is called the remainder or the dif"
/erence.
Ex. From 794327 In order to suhtract one whole number from
Take 342814 another, we first place the number to be sub-
451513 tracted under the other, with their units-figures
in the same line ; we then take the units-figure, 4, of the lower number
from that of the other, 7, thus 4 from 7, 3, i. e. 3 units, and we place the
3 under the units-figures, to be the units-figure of the result ; then we
proceed to the tens-figures, and say, 1 from 2, 1, L e. 1 ten, and we set
down I under the tens-figures; then to the hundreds-figures, and say 8
from 3. . . / cannot; but if we take or borrow 1 out of the 4 tliousands
(leaving 3 thousands), and treat it as 1 ten of hundreds, we shall now
have 13 hundreds in the upper line; we can now say 8 from 13, 5, i. e.
5 hundreds, and we set down 5 as the hundreds-figure of the result : and
we have now to take 2 thousands from 3 thousands, or, which is just the
same, but more convenient in practice, instead of supposing the upper
figure, 4, diminished when we borrow 1, we may suppose the lower cor-
responding figure, 2, increased, i. e. we may carrj/ one to it, and say 3
from 4, 1, i. e. 1 thousand, and so on.
N. B. Any sums may be set at pleasure in Subtraction, and the
Answers proved by adding the remainder to the bwer number, when
the result will be the upper, if the sum be worked correctly.
EXAMPLES IN SUBTRACTION.
1. 765439 2. 697438 3. 758452 4. 543625
343418 635036 418234 492708
- ■ t
5. 683125 6. 712345 7. 564307 8. 702306
492816 538159 479176 475429
9. F/om six hundred and nine thousand seven hundred and one take
three hundred and ninety-seYQU thousand and forty-cine.
SIMPLE MULTIPLICATION. 5
10. From four hundred and fifty thousand and ninety-four take ninety-
nine thousand nine hundred and nine.
11. From seven hundred and eighteen millions fourteen thousand and
fifty -seven take ninety-seven millions eight hundred and four thousand
seven hundred and sixteen.
12. From fifty-three thousand millions eighteen thousand and ninety-
seven take forty thousand five hundred and twenty-eight millions seven
hundred and six thousand seven hundred and nine.
Multiplication is the method of finding what number would
result from adding several of the same numbers together ;
thus, if we add 6 sevens together, the result is
7-1-7 + 7+7 + 7+7=42,
♦he same number as that given in the Multiplicatjon-table
for the value of 6 times 7 : and, since the same number is
also the sum of 7 sixes^ or the value of 7 times 6, it follows
that, when two numbers are multiplied together, it matters
not which we take as multiplier.
The numbers multiplied in any case are called /ac/or^, and
the result is called the product,
Ex. 1. 3467 When the multiplier^ as in Ex. 1., is not higher than
2 12, we first set it with the units-figure under that of
6934 the multiplicand; then we hegin to multiply, saying,
twice 7 is X^—four and carry one^ i. e. we set down the 4 units under the
units- figures, and carry the 1, which means 1 ton, to be added to the
tens; we now proceed, twice 6 is 12 (i.e. 12 tens^ since 6 means 6 tens),
and 1 (i. e. the one carried) is\3.,,3 and carry 1, i. e. we set down the 3
tens, and carry the 1 , which means 1 ten of tens or 1 hundred, to be added
to the hundreds, and so on throughout the line.
Ex. 2 3467. ...2 When the multiplier, as in Ex. 2., is higher
692 . ...8 than 12, we first set it under the multiplicand
6934 as before, and, having multiplied the upper
31203 line by the units-figure, 2, of the lower, as in
2^^^^ Ex. 1., we now multiply by the tens-figure, 9,
2399 164.. ..7 saying 9 times 7 is 63 (i.e. 63 tens, since 9
means 9 tens) ... 3 and carry 6; i. e. we set down the 3 tens, and carry
the 6 tens of tens or hundreds, and so on: we now multiply by the
hundreds-figure, 6, of the lower line, in the same matvww, «xv\ \X\«cl ^^^
'ttP i^** s^p^T^ate lines, when the result is iVie pxo^McX. x^o^vt^^ ^\i^
^8, and 7, on the rif^ht, will be ei^plsdned i^t^qt\X\j.
6 SIMPLE MULTIPLICATION.
Ex. 3. 37218 Since it is immaterial which number we take as
^ multiplier, it is best always to choose that which is
223308 simplest; and if it can be separated into two or more
I factors each less than 12 (thus 42=^6 x 7), we ma/
1563156 multiply separately by each, as in Ex. 3.
N.B. Any number which can be separated into factors is called a
composite number; any number which cannot be so separated, such as
7, 11, 13, 17, &c., is called & prime number.
^' *' ^^2700 ^^ *^® multiplier ends with one or
— ^ — - — ' * * more cyphers, the sum may be worked
gjg^ as in the annexed example, by which
'TZTZTZx r. many useless cyphers are saved.
o7oOiOO O
N. B. Any sums may be set at pleasure in Multiplication, and the
Answers proved, either by repeating the operation with the other number
for multiplier; or by the process of casting out nines (for the proof of
which see Algebra), as follows: add up the figures in the upper number,
divide this by 9, and set down the rem*"; do the same with the other
number; then do the same with the product of these rem'*, and with the
product of the two numbers; and if the new rem" arc the same, the sum
is most probably right; but, if different, it is pertainly wrong. Thus in
Ex. 2., the first pair of rem" are 2 and 8, and their product 16; the
rem' from this is 7, the same as from the Ans': in Ex, 4., the first pair
of rem" are 1 and 0, and their product is 0; the rem' from this is 0, the
same as from the Ans'. See Note I.
It is desirable that the pupil should be made to apply one or both of
these methods to the Examples below given. /
EXAMPLES IN MULTIPLIC^Tie^.
1. 345673x2 2. 457632x3 " 3. 415763x4
4. 371281x5 5. 635432x6 0. 421375x7
7. 378914 X 8 8. 476539 x 8 9. 435976 x 9
10. 978564x11 11. 496782x12 12. 876549x12
13. 378125x16 U. 456932x18 15. 712436x24
16. 543817x27 17. 593654x30 18. 697128x36
19. 765438x40 20. 596437x45 21. 642198x60
22. 756328x72 23. 814765x84 '24. 913748x96
25. 234915x123 26. 704745x615 27. 469830x369
28. 391525x861 29. 1174575 x?214 30.3523725x2583
31.1644405x7749 32. 231549x8856 33. 463098x7380
34. 1389294x8900 35. 926196x7896 -^6, 2778588x9867
' ■' - >
SIMPLE DIVISION. 7
Division is the metliod of finding how often one number is
contained in another, i. e. how often one number must be
taken to make up another. Hence Division bears the same
reference to Subtraction, as Multiplication bears to Addition;
for we might go on subtracting the divisor from the divi-
dend, and then from the 1st rem'', then from the 2nd rem'',
and so on, until the final rem' is either zero, or is less than
the divisor itself; and if we counted the number of times we
had subtracted it, this would be the result required, or, as it
is called, the quotient. But the Multiplication -table will
enable us much more easily to divide one number by another;
thus, since 7 times 9 is 63, if we divide 63 by 7, we shall have
the quotient 9, or if by 9, the quotient 7 : and the method of
applying it to more difficult cases will be seen by what follows.
Ex. 1. 4) 2379 When the divisor, as in Ex. 1., is not higher
— TTT^ than 12, we first set it in a loop before the
^ dividend ; then we take the first figure of the '
dividend, 2, i. e. 2 thousands : but, since 4 will not be contained at all
in this, wc take then the first two figures, 23, i. e. 23 hundreds, and say
4 is in 23 . . 5 times and 3 over, and we set down the 5, i. e. 5 hundreds,
m the quotient, and carry the 3 hundreds, or 30 tens, to the tens-figure,
7, of the dividend: we have now 37 tens, to be divided by 4; we say,
therefore, 4 is in 37 . . 9 times, and 1 over, and we set down the 9, i. e.
9 tens, in the quotient, and carry the 1 ten or 10 units to the units-
figure, 9, of the dividend: we have now 19 units to be divided by 4;
we say, therefore, 4 w in 19 4 times and 3 over, and we set down the
4, 1 e. 4 units, in the quotient, and place, as is usual, the final rem*
3 over the divisor with a line between them, as 5 (three-fourths), &quAniity
meaning 3 -r 4, and called a fraction, of which more will be said hereafter.
It appears then that 4 will be contained 594 times in 2379, with 3
over; i. e. we might subtract 4 from 2379 594 times, and have still 3
remaining. This is an example in Short Division,
Ex. 2. 42) 379543 (9036JJ When the divisor, as in Ex. 2., is
- ^8 Ivgher than 12, we place it, as before, in
154 a loop before the dividend, and the quo-
126 tient in a loop nftcr it ; and we see that
283 42 will not be contained in the 3 (i. e. 3
252 hundreds of thousands)^ ivox V^ xXvi. ^1
31 (i. c. 37 tens of ihousauds^, \svx\.^*^\ ^^
8 SIMPLE DIVISION.
contained 9 times in the 379 (i. e. 379 thousands)', or, which is the same
thing, but more convenient in practice, we take the first figure only of
the dividend, and say 4 is in 37 . . 9 times; we set therefore the 9 (i. c. 9
thousands) in the quotient, and, multiplying 42 by 9, subtract the pro-
duct, 378 (i. e. 378 thousands) from the dividend; and we have now the
rem', i. e. 1 thousand or 10 hundreds, to be carried to the hundreds: we
i
/ take in then the hundreds-figure, 5, of the dividend, and have now 15
' hundreds to be divided by 42; we say then (42 is in 15, or) 4 is in 1 . .
I cannot; we set, therefore, (i.e. hundreds) in the hundreds place of
the quotient, and have now 1 5 hundreds, or 1 90 tens, to be carried to the
tens; we take in then the tens-figure, 4, of the dividend, and have now
154 tens to be divided by 42; we say then 4 is t'/i 15 . . 3, and we set the
3, i. e. 3 tens, in the quotient, and so on till, at last, we have the final
rem' 31, which we set over the divisor, as a fraction, and have the whole
quotient 9036^. This is an example in Long Division.
Ex. 3. Bat when the divisor, as in this case, is made np of two or
more factors, less than 12, it is often more convenient to divide by each
separately, as follows.
6 ) 379543 There is here a fraction | over in the first quo-
7) 63257i tient, and a rem' 5j in the second, which, according
9036U to our previous practice, should be written -^ ; but
such an expression may always be simplified (as will be shown here-
after) by putting the rem' 5| in the form ^, (which we obtain by multi-
plying the 5 by the 6, and adding in the 1); and then multiplying the
6 by the 7, so making ^, the same as the fraction obtained by the other
method. See Note IL
Ex. 4. 39,00) 7134,53 (182|§§§ In this Ex. and in all others where
39 there are cyphers at the end of the
323 divisor, the work may be abridged
312 by marking off, with a comma, or
114 point, these cyphers, and as many
"8 figures also from the right of the
3653 dividend ; then we proceed, 3 is in
7 twice; but on trial we should find that 2 would be too larg*
first figure in the quotient, (which comes of using 3 for the divisr
of 39, and this difficulty will sometimes occur, but not so as to.
the student, when he gets accustomed to division); we set, there
the first figure in the qnoticnt, and go on, as before, till we have «,
doirn aU the iiffurcs before the point in the dividend ; and then we com-
ANSWEBS TO THE EXAMPLES. 9
picte the last rem' by taking down the two figures cut off, and put it over
the divitior as a fraction.
N. B. Any sums may be set it pleasure in Division, and the an-
swers proved hy either of the methods given in Multiplication ; since
the product of the divisor and quotient (if the sum be worked correctly)
will give the dividend, diminished, however, by the remainder (or upper
number of the fraction; if any. Thus in Ex. 2 , the divisor is 42 and
quotient 9036, and the rem" from these are 6 and 0; the product of
these is 0, and the dividend, diminished by the rem' 31, is .379512, and
the rem'» from these are 0, : in Ex. 4., the di'-isor is 3900 and the
quotient 182, and the rem'" from these are 3 and 2 ; the product of
these is 6, and the dividend diminished by the rem' 3653, is 709800,
and the rem'" from these are 6, 6.
The pupil should be required to apply one or other of these methods
of proof in the following examplos.
1. 432516 -^ 2.
4. 7I3915-^5.
7. 465328-5-8.
. 10. 457848 -i- 11.
13. 2366745-5-15.
16. 6549372 -^ 36.
19. 7825687-5-64.
122. 6.598769-7-84.
EXAMPLES IN DIVISION.
2. 351 789 -^ 3. 3.
5. 385734-5-6. - 6.
8. 395424-^8. 9.
11. 716855 -r 12. 12.
/
^4. 7954326-7-18.
17. 473349^45.
20. 3795469 -r 70.
-f-23. 8791605-f-88.
543756 -r 4.
616824+7.
567035-^9.
936571 -T- 12.
15. 63425764-24.
18. 5674331-5-60.
2L 8754329 -f 80.
24. 7654325 -s-96.
25. 3765897-^23. 26. 4613578-*.3?. 27. 5123495-f41.
28. 3954371-^47. 29. 3755123-5-234. 30. 5764123-i-84a
31. 34568135-5-357. 32. 76549139-5-543. I'JfiS. 29876533-5-6930
34. 56854327-5-7323. 35. 95642371-5-8790. 36. 34568795 -f 987a
~ *
ANSWERS TO THE PRECEDING EXAMPLEa
ADDITION.
/ 1274170.
2.
1634607.
3.
1659291. 4. 2333331.
/ '3005313.
6.
1537206.
7.
1648127. 8. aO^I^Wl
9. 1835161.
10.
ll^^^l^.
U. 22m29927'
\2.
205^^1^^^^^
\
10
ANSWERS TO THE EXAMPLES.
SUBTRACTION.
1.
422021
2.
62402.
3.
340218.
4.
50917.
6.
190309.
6.
174186.
7.
85131,
8.
226877.
9.
212652,
10.
350185.
11.
620209341.
/
12.
12471?llp88.
691346. /
/
MULTIPLICATION.
1.
2.
1372896.
3.
16C3052.
4.
1856405.
5.
3812592.
6.
2949625.
7.
3031312.
8.
3812312.
»•
3923784.
6050000.
10.
14.
10764204.
11.
5961384.
12.
16.
10518588,
13.
8224776.
15.
1 7098464.
14683059.
17.
17809620.
18.
25096608.
19.
30617520.
20.
26839665.
21.
38531880.
22.
54455616.
23.
68440^60.
24.
87719808.
25. 28894545.
28. 337103025.
31. 12742494345.
^34. 12364716600.
26. 433418175.
29. 2p00p09,050.
32. 2050597944.
. 35. 73132^43616.
27. 173367270.
30. 9101781675.
fe3. 3417663240.
36. 27416327796.
1. 216258.
5. 64289.
9. 63003|.
13. 157783.
16. 181927.
NL9. 122276|f.
^82. 78556^.
DIVISION.
2. 117263. a '135939.
6. 73832. 7* 58166.
10. 41622iL. if, 59737ii.
14. 441907.
17. 10518851.
20. 64220?g.
^ 23. 99904^5.
4.
8.
12.
142783.
49428.
78047^.
15. 264274,
^8. 945721-J.
21. 46929^.
24. 79732^.
163734lf.
25.
28.
31.
34. 7763^^
84135|?.
96829|ff.
587
732
26.
29.
32.
35.
124691J^.
160471M.
140974f|i.
1^60|^
27.
30.
33.
36.
124963|f.
169531^.
/
•Tl_.
P]
^« = = = -.=-« =
= «
^c^„,.«,a.oc
;:s
•diSSSSSSSSSJS^
-a
"-i
•d^SSPSSSa
5gS
s
g
5^2 ali'a.'s
aiifia
^^^ s't^ITi'^
S-'t!§i''
lili
-2s -al-sl
",
n
^
s
3
s
k|3
s
slsi
^
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a
s
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s
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'
3
.
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o.
.
£:
s
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^j«* ae «7Qjao ce m -« ^ 19
^^w
12 REDtrCTION.
Reduce Ex. 1.
1. £513 to farthings ; and 320 guineas to halfpence.
2. £2000 to halfcrowns ; and 2000 guineas to sixpences.
*— 3. £27 10*, to pence ; and 17«. G^d, to farthings.
^ 4. £75 105, 6d. to sixpences ; and 220 crowns to fourpcnny-picccs.
"5. £47 10*. llfrf. to farthings, and £85 0*. lO^d. to halfpence.
6. £29 10*. Oirf. to halfpence ; and 1373 halfcrowns to farthings.
7. 23 tons to pounds ; and 115 cwt to ounces.
8. 27 Ihs. to drams ; and 1 1 tons to ounces.
^'9. S qrs. 14 oz. to drams ; and 47 cwt. 25 lbs. to ounces.
10. 34 cwt. 3 qrs. 1 1 oz. to drams ; and 2 tons 3 qrs. 5 oz. to ounces.
11. 4 tons 15 cwt. 2 qrs. 12 lbs. to lbs. ; and 14 cwt. 1 qr. 8 drs. to drams.
"-12. 15 cwt. 2 lbs. 9 oz. to ounces ; and 3 tons 3 qrs. 3 oz. to drams.
13. 16 lbs. Troy to grains ; and 105 lbs. Troj to dwts.
14. 27 oz. 10 dwts. to grains ; and 3 lbs. 13 dwts. to dwts.
15. 9 oz. 17 dwts. 22 grs. to grains ; and 2 lbs. 11 oz. 20 grs. to grains.
^6. 7 oz. 19 dwts. to grains ; and 3 lbs. 9 oz. 7 grs. to grains.
1 7. 23 miles 7 fur. to feet ; and 2 lea. 2 m. 7 fur. to yards.
. 18. 3 fur. 135 yds. 4 in. to inches ; and 5 fur. 171 yds. 2 ft. to inches.
. 19. 2 lea. 2 m. 2 fur. 200 yds. to feet ; and 5 m. 200 yds. 3 in. to inches.
20. 73 yds. 3 qrs. to nails ; and 35 ells 4 qrs. to nails.
21. 54 A. 3 R. to poles ; and 17 sq. yds. 8 ft. to inches.
22. 7 A. 12 p. to poles ; and 29 sq.yds. to square inches.
23. 13 cub. yds. to feet ; and 7 cub. yds. 20 ft. to inches.
24. 23 cub. yds. 1000 in. to inches ; and 12 cub. yds. 23 ft. to inches.
25. 137 gals, to pints ; and 13 gals. 3 qts. to gills.
26. 17 qrs. to gals. ; and 220 bushels to quarts.
27. 3 loads 3 qrs. 3 pks. to gals. ; and 2 qrs. 1 gal. to pints.
28. 3 loads 3 bus. to quarts ; and 2 qrs. 7 bus. 2 pks. to gallons.
29. 27 years to days ; and 3 yrs. 315 d. to minutes.
30. 5 mo. 3 w. 4 d. to hours ; and 27 w. 5 d. 15 hrs. to seconds.
3. To reduce a quantity to a hiy*^v, wV;:::^.:
Rule. Divide the given quantity by the number wliich
; . shows how many of the lower denomination make one of the
I next higher ; and so on, step by step, till we arrive at the
proposed higher denomination.
Ex. 1. Reduce IS7 520 farthings to shillings.
4) 137520/1 Here we first divide the given number of far-
,gN oioon^ things by 4 to bring theni into pence, and then
we divide these pence by 12 to bring them into
REDUCTION. 13
If there should be a remainder after any division, we must
set it down as a term of the same denomination as the
dividend from which it came.
Ex. 2. Beduce 13799 farthings to pounds.
4) 13799/. Here, after dividing the given farthings by 4,
12) 3449rf . 3/. ^® ^*^® * ^^^' ^» which means that in 13799/
.-— -T— there are 3449</., and 3/ over; we set down
* ' — 1—L ' ' ' ' therefore the rem' as 3/1, that is, as a term of the
£14 7*. 5jrt. Ans, ssLme den" as the dividend from which it came ;
after dividing the pence bj 12, we have a rem' 5, which we set down, for
a similar reason, as 5d, ; and after dividing the shillings by 20, we have
a rem' 7, which we set down as 7*.
N.R We have divided by 20 by the usual short method, cutting off
the last figures of the dividend and divisor.
Beduce Ex. 2.
1. 787902365. to guineas ; and 150080 sixpences to pounds.
2. 1758960/1 to crowns ; and as many halfpence to halfcrowns.
3. 480144/ to sevenshilling-pieces ; and 50000(/. to pounds.
4. 284061/ to pounds; and 110012dL to pounds.
5. lOlOlOef. to guineas ; and 123290/*. to pounds.
6. 350000/1 to pounds ; and 538483 halfpence to guineas.
7. 37568 lbs. to tons ; and 108190 drs. to cwt.
8. 2345820 drs. to tons ; and 108234 oz. to cwt
9. 100000 oz. to tons ; and 12821 drs. to qrs.
10. 229601 oz. to tons ; and 314735 drs. to cwt.
11. 156423 drs. to cwt. ; and 1008001 oz. to tons.
12. 237023 oz. to tons ; and 371283 drs. to cwt.
18. 13172 grs, to lbs. Troy ; and 30066 dwts. to lbs. Troy.
14. 17073 grs. to lbs. ; and 12327 grs. to lbs.
15. 108970 grs. to lbs. ; and 189081 grs. to lbs.
16. 272821 grs. to lbs. Troy ; and 127272 grs. to lbs. Troy.
17. 36090 ft. to miles ; and 231031 yds. to leagues.
18. 120835 in. to furlongs ; and 378 136 ft. to miles.
19. 51 7900 in. to miles ; and 183810 ft to leagues.
20. 13587 na. to yards ; and 181970 na. to ells.
21. 121321 p. to acres ; and 33333 sq. inches to yards.
22. 20000 p. to acres ; and 20000 sq. inchcs-to yards.
23. SOOOOO cab. in. to yards ; and 138297 cub. in. to yards^
24. 106921 cub. in. to yards ; and 180831 cub. in. to yard<&.
2{^. I8I9I pts. to gallons ; and 30983 glWs to ^a\\oxk&.
14 COMPOUND ADDITION,
26. 28716 qts. to loads ; and 91356 pints to quarters.
27. 89765 pks. to loads ; and 56789 pts. to loads.
28. 356187 qts. to loads ; and 598712 gals, to quarters.
29. 137819 days to years ; and 3561829 sec. to weeks.
30. 235967 hrs. to weeks ; and 71871900 sec. to years.
Compound Addition,
4. Rule. Set the quantities to be added under one another,
<o that terms of the same kind may be in the same column.
Add the numbers in the right-hand column ; divide the
result by the number of things in this column, which make
one in the next ; set the remainder, if any, under the first
column, and carry the quotient to be added to the next ; and
so on with all the columns.
£ 8, d.
Ex. 1. 13 8 Here, adding up the pence in the right-hand
2 5 6 column, we have A2d, ; in order to bring this into
23 4 7 shillings, we divide by 12, which goes 3 times with
37 8 10 6 Qyer^ go that 42e/. = 3s. 6</. ; we set down the 6ci
under the first column, and carry the 3*. to tlio
next ; and so on.
r
Ex. 2. 22 4 6j Here, adding up the farthings in the right-hand
2 65 column, we have if., which«l|(/. ; we therefore
36 4f set down the |(/., and carry \d. to the next column.
7 1 li
12 S 7
13 4
£89 2 6
£ 8, d.
£65 8 6J
Ex. 3.
£ 8, d, £ 8. d, £ 8, d, £ s. d,
1. 3 13 6 2. 14 13 7 3. 65 4 3| 4. 23 13 6|
2 11 9 22 15 9 22 2^ 35 17 OJ
3 17 8 29 11 11 46 15 7* S5 7 r{
2 5 2 82 17 7 73 12 6? 67 16 ^
•i
5. 41 16 8J 6. 36 17 65 7. 24 16 8^ 8. 71 17 21
21 10 7i 14 17 6 61 14 2] 41 2 91
31 17 7J 21 12 72 11 8 54 7 6^
24 16 81 13 13 31 27 1 3 2 116
9. 16 5 4 10. 11 13 3J 11. 42 13 4 12. 76 15 45
35 7 9j 32 12 21 17 6 8^ 32 4 10
16 10 8 13 13 3| 90 9 8 21 3 71
42 33 8| 24 3 21 12 4i- 62 18 4^
COMPOUND ADDITION. 15
lb. OS. dr. qr. lb. oz. cwt. qr. lb. qr. lb. oz.
n. 7 3 13 14. 3 27 15 15. 18 2 23 16. 13 25 7
12 9 1 11 2 17 1 19 4 18 6
23 13 14 21 13 15 3 17 24 17 5
3 15 7 2 13 14 9 2 25 37 9 U
qr. lb. oz. dr. cwt. qr. lb. oz. tons cwt. qr. lb.
17. 2 15 13 11 18. 27 2 ' 13 4 19. 4 17 8 18
3 5 11 8 32 1 12 15 2 3 15
2 27 13 2 28 15 12 13 9 2 25
3 17 15 4 32 1 14 3 22 18 3 15
oz. dwt.
gr.
lb.
oz. dwt.
02, dwt
gr.
lb.
oz. dwt.
20.
9 17
23
21.
23
8 14
22. 7 17
21
23.
25
8 14
4 IS
20
7
9 19
11 5
13
37
3 15
7 5
15
37
5 3
4 14
20
25
9 10
8 19
4
gr.
15
7 13
oz.
10 17
5
lb.
44
oz.
7 11
lb. oz.
dwt.
lb.
dwt. gr.
dwt gr.
24.
12 5
13
22
(
25, 35
3
4 12
26.
27
17 22
24 7
19
13
27
8
14 22
5
9
23
47 11
17
19
41
9
17 10
17
8
11 13
31 4
11
17
oz
2
3
13 21
gr.
22
7
9 15
dr. 8cr.
gr.
%
dr. scr.
dr. scr.
oz.
. dr. scr.
27.
5
13
28.
11
7 2
29. 7 1
19
30. 11
7 2
7 2
14
4
3 2
8
I
10
5 2
3 1
17
10
5 e
11 2
13
5
2 I
6
12
9
4 1
9 1
14
11
6 2
yds. ft
in.
fur.
po. yds.
m. fur.
yds.
lea
. m. fur.
31.
12 1
11
32.
. 7
31 41
33. 5 7
137
34. 7
1 6
22 2
9
3
19 2J
2 4
121
8
2 4
9
3
8
27 3
8 6
213
1
5
13 1
4
4
35 5
3 5
23
9
po.
1 7
fur. po.
yds.
po.
yds. ft.
yds. ft.
'in.
yds. in.
35.
5 33
^
36.
27
^ 2
37. 5 2
10
38.
, 7
31 11
7 21
H
35
H 1
8 1
4
9
2 10
2 13
H
24
41
6
7
5
11 8
6 21
5
In.
13
3 1
ur.
9 2
5
m.
6
2^ 6
po. j-d8<
, ft.
m. f
po. yds.
fur.
yds. in.
39.
7 3
1
11
40. 14
3
17 2|
41
. 3
5
137 9
12 2J
2
4
23
5
33 4
7
7
77 7
9 4
7
37
1
24 5
9
6
2Qa ^
,
2 31
1
9
43
7
31 11
^
\
X^'ci "^
G2
16 COMPOUND ADDITION.
yds. qrs. na.
yds. qrs. na.
ells qrs.
na.
ells qrs. na.
42.
25 3 2
43.
188 3 2
d
14. 79 3
3
45. 35 2 3
37 3
297 1
67 4
1
42 4 5
54 1 1
328 2 3
82 I
3
37 2 2
49 2 3
169 1 2
98 3
2
25 4 3
1
B.ydi. i.ft. s.in.
R. P. 8. yds.
t
A. R.
p.
A. R. P.
46.
20 8 100
47.
7 33 20i
48. 27 2
31
49. 27 1 31
31 7 85
8 13 14l
35 3
24
41 2 28
24 5 34
7 25 'J^
22 1
17
51 19
S7 8 113
p. 8. ydi. 8. ft.
, 8. in.
6 17 11
p.
45
s. yds.
29
R.
42 1 25
A. R.
p. 8. yds. 8. in.
50.
2 13 7
85
51. 35 1
23
12i 5!
2. 37
33 23i 121
3 20} 8
24
9 2
15
27i
21
25 17 135
5 25| 6
99
.11 1
24
11
18
17 20i 102
4 225 8
c.yds. eft. c.
37
in.
42
c. yds.
35
c.n
:. c.in.
25
12 25 97
i
c. yds. c. (I. c. in.
53.
13 25 872
54. 27
22
856
55
. 14 20 1431
22 17 1000
31
15
979
32 3 1560
34 11 1534
24
19
787
25 18 937
21 8 479
22
6
842
22 21 1364
gal. qt8. pts.
gal. qts. pts.
pks. gal. qts.
bus. pks. gal.
56.
27 3 1
57. 17 3 1
58. 3 1
3
59. 23 3 1
31 2
24 2 1
4
2
31 2 1
54 1 1
35 3
5 1
I
24
37 1
25 2 1
7 1
3
35 3 1
qrs. bus. pks.
Ids. qri. bus.
bus. g.i1
1. qt9.
bus. pks gal.
60.
13 3 2
61. 13 4 7
62. 31 1
3
63. 29 3 1
24 6 1
24 3 4
25
2
37 2
37 3 1
37 4
41 1
1
53 3 1
43 5 2
43 2 1
27 I
3
47 2 1
gal. qts. pts. 1
gills.
bus.
pks.
gal. qts.
qrs. bus. pks. gal.
64.
22 3 1
3
65. 13
2
1 3
66. 23 3 3 1
31 2
1
42
3
1 2
32 4 I
13 3 1
2
51
1
3
41 6 2 1
24 3 1
1
47
3
I 2
52 2 1
d. hrs. min.
, sec.
mo.
w.
d. lirs.
d. hrs. min. sec.
67.
5 13 39
42
68. 13
3
5 11
69.
4 11 39 28
4 22 19
33
21
2
4 15
2 13 10 32
6 20 29
45
37
3
6 17
5 21 40 29
4 17 59
69
41
2
5 19
7 23 19 19
yrs. d. hrs. min. yrs. w. d. hrs. yrs. d. hrs. min.
70. 6 130 23 15 71. 14 13 5 23 72. 8 244 22 49
7 354 10 17 22 47 4 3 6 315 17 38
S 45 22 14 35 39 3 18 5 223 13 45
9 97,9 13 17 21 44 6 15 1 \'i^ •i\ ^%
£ 8,
Ex. 1. 34 17
27 8
d.
9f
4|
£7 9
£ 8.
Ex. 2. 19 12
16 17
5J
*
COMPOUND SUBTBACTIOK. 17
Compound Subtraction,
5. Rule. Set the quantity to be subtracted under the
other, BO that terms of the same kind may be in the same
column.
Subtract the right-hand term of the lower line from that
of the upper, if possible ; if not, subtract it from the number
of things in this column, which make one of those in the
next, and add the upper term to the remainder ; place the
result under the first column, and carry one thing to the
lower term of the next ; and so on with all the columns.
Here, taking |<f. from \d., we have left Jdl to he
set under the farthings ; then taking Ad, from 9^/.,
we have left 5d, to he set under the pence ; and so
on.
V
Here we cannot take |rf. from Jrf. ; we borrow
threfore \d, from the G.i, and convert it into far.
things, thus changing the Sjc?. into 7c?. + 1^(1, or
£2 15 3f 7d. 6/; taking, then, the \d. or 2/ from 5/, we
have left 3/« or |c?., to he set under the farthings, and have now to take
Ad. from 7</., which leaves Srf. to be set under the pence.
N. B. In practice, it is best to take the |c?. at once from the \d, bor-
rowed, which leaves |dl, and add in the \i. to this rem', which gives |<f,
as before; and also, instead of taking Ad. from 7(f., we may take 5d. from
8(f., which will leave the same rem' 3(f., t. e, we need not alter the quan-
tity from which we subtract, if we add, or carry^ one to the quantity
subtracted.
Again, as we cannot take 11 8. from 12*., we borrow £1 from the £19,
and thus taking lis. from £\ \2s, or 32«., we have left 15«., and then,
taking £16 from £18, we have left £2. Here, too, it is best to take the
17*. at once from the £l borrowed, which leaves 35., and add to this the
12«., which gives 15*. as before; also, to carry £1 to the £16, making
£17, and take this from the original £19, which leaves £2 as before.
£ ,, ^ Here, taking \d. from \d. borrowed, we have JdL
Ex. 3. 23 6 Oj left, to which we add the \d,^ making \d, to be set
2 2 18 11 | ^own; then carrying Id, to the ll</.,we have 12<i,
£0 7 0| which we take from Is. borrowed^ and have no rem'}
again, carrying 1*. to the 18*., we have 19*., wliich we take from £l
borrowed^ and have 1*. left, to which we add the 6*., making 7*. to be
set down; and carrying £l to the £22, we have £23 to sabU^jt^X, %sw^
no rem'.
18 COMPOUND SUBTEACTION.
Ex. ft.
1.
£ s. a,
23 10 8
2.
£ s. d.
45 14 7i
3.
£ s. d.
74 6|
£ s. d,
4. 89 15 7
13 7 5
12 7 5i
13 8 4.1
74 11 9
5.
93 9
6.
24 5
7.
132 11 61
8. 225
•
37 10 11
15 12 11
129 13 41
37 18 9|
9.
137 13 OJ
10.234 llj
11.
317 14 01
12. 345
111 15 92
195 18 102
239 18 lOf
129 17 82
13.
lbs. oz. dr.
27 11 3
14.
qrs. lbs. oz.
13 3 1
15.
cwt. qrs. lbs.
33 11
qrs. lbs. oz.
16. 2 23
13 7 1
5 12 14
12 1 24
1 25 9
17.
qrs. lbs. oz.
17 11 3
18.
tons cwt qrs.
32 1 1
19.
cwt. qrs. oz.
27 I 3
cwt. lbs. oz.
20. 45 3
8 27 15
30 14 3
13 7
44 6 13
21.
oz. dwt. gr.
11 19 3
£2.
oz. dwt gr.
32 7 21
23.
lbs. oz. dwt.
13 7 15
oz. dwt gr.
24. 11
8 14 17
18 9 22
6 n 18
2 18 22
25.
oz. dwt. gr.
23 4
26.
oz. dwt. gr.
37
27.
oz. dwt. gr.
22 2 2
oz. dwt. gr.
28. 42 3
1 15 20
11 13
13 11 11
27 13 21
29.
dr. scr. gr.
7 1 18
30.
oz. dr. scr.
11
31.
lbs. oz. dr.
37 7 1
dr. scr. gr.
82. 8 11
4 19
8 5 2
19 11 2
6 2 15
S3.
yds. ft in.
13 1 7
34.
po. yds. ft.
23 3 1
35.
fur. po. yds.
6 37 2
m. fur. yds.
36. 13 6 123
11 2 10
13 41 2
1 15 41
8 7 219
37.
m. fur. po.
24 7
38.
fur. po. yds.
6 37 4
39.
lea. m. fur.
37 5
fur. po. yds.
40. 7 23 31
11 6 18
5 18 41
18 7
6 35 5
41.
po. yds. ft.
23 3 2
42.
yds. ft. in.
23
43.
yds. qrs. na.
17 3 2
ells qrs. na.
44. 24 1 3
15 41 1
15 2 7
i3 1
19 2 1
45.
s.yds. 8. ft. s.in.
13 2 73
46.
p. s.vds. s.ft.
22 13 5
47.
R. P. s.yds.
3 2 25
A. R. P.
48. 37 2 29
6 8 131
13 201 8
2 35 281
23 3 35
49.
A. R. P.
45 2 35
50.
R. P. B.yds.
2 35 20
51.
R. s.rds. s.ft.
10 131 4
s.yds. s.ft. s.in.
52. 12 2 13
19 3 39
1 21 281
8 10 7
8 7 130
COMPOOND MULTIPLICATION.
19
c,yds.c.fl. cin. c.yds.c.ft. cin. c.ydS'C.ft. c.in.
53. 23 13 357 54. 37 2 459 55. 45 24 656
10 25 1014 7 24 1532 12 19 999
pks. gals. qts.
59. 23 1
19 1 3
bus. pks. gnls.
63. 57 1
39 3 1
w. d. hrs.
67. 3 5 2
2 6 13
yrs. w. d.
71. 45 45 3
35 1 6
57.
gals. qts. pts.
36 2
33 3 1
61.
qrs. bus. pks.
45 3 1
39 7 2
65.
hrs. m, B.
22 39 19
8 41 30
69.
yrs. d. hrs.
32 131 22
19 300 13
58.
gals. qls. pts.
35 1
29 3
62.
Ids. qrs. bus.
22 3 5
9 3 7
66.
d. hrs. m.
14 17 20
6 21 35
70.
yrs. w. d.
27 35 4
18 47 6
c.yds.c.ft. c.in.
56. 27 13 2
13 23 731
bus. pks. gal.
60. 47 2
28 3 1
Idi.qrs.bus.
64. 5 11
2 4 5
mo. w. d.
68. 12 2 5
8 3 6
yrs. d. hrs.
72. 26 213 11
19 231 21
Compound Multiplication,
6. EuLE. Set the multiplier under the right-hand term
of the multiplicand ; multiply this term by it, and find, as
before, how many are to be carried to the next term, writing
the rem'^ under the right-hand term : then multiply the next
term, and add in the number carried ; and so on.
Ex. !. £23 13
5
4
Ans. £94 13 8
Ex. 2. £37 13
11
8J
^RS. £414 10
9i
Ex. 8.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
£
23
59
78
99
171
134
165
115
124
171
37
128
s.
8
13
2
17
13
6
14
7
5
13
17
d.
4
7
8
5
2
9
2
9
4
11
Here 5d, x 4—20cl. = 1«. 8flf. ; we set down 8(f.,
and carry 1*. : — 13*. x 4=525., and, adding the
1*. carried, we have 53*. =£2 13*.; we set down
13*. and carry £2:— £23 x 4=£92, and, adding
the £2 carried, we have £94.
Here 2/. x ll=22/.=5rf. 2/ or b^dr, we set
down |(/., and carry 5d. ; and so on.
£ 8. d.
37 13 5ix
48 17 7ix
2
3
4
5
6
7
8
9
xlO
xU
25 xl2
3 xl2
X
X
X
X
X
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
22.
24.
96 15
75 14
154 11
161 12
173
135
175
183
51
6|x
2|x
32 X
7jx
5ix
4|x
2
3
4
5
6
7
8
9
18
15
4
12 lOfxU
10 0jxl2
9|xlO
n\ \^ >V^>^
?o
COMPOUND MDI.TIPLICATION.
I*
When the multiplier is large, but is composed of two or
three factors*, we may multiply separately by each of these.
Ex. 1. Multiply £23 lU. 4|d by 36.
Since 36=^6 x 6, or=4 x 9, or=3 x 12, the sum may stand thus:
£
8,
d.
£
s.
d.
£
8.
d.
23
11
4f
6
or 23
11
4f
4
or
23
11
4
12
141
4^
94
7
9
282 16
848 10 3 Ans, 848 10 3 Ans,
Ex. 2. Multiply £\1 3». Ojrf. by 140.
Since 140 = 4 x 5x7, the sum may stand thus;
£ 8, it.
17 3 01
4
9
3
848 10 3 Jns.
68
12
2
5
343
10
7
2401 5 10
Ex.e
£ 8.
d.
£ 8,
(t.
1.
23 17
5ix 15
2.
79 14
lOjx 18
3.
93 8
3ix 21
4.
49 12
8 X 28
5.
68 7
4|x 35
6.
97 19
91 X 48
7.
87 4
3jx 64
8.
92 11
10 X 70
9.
37 13
2ix 81
10.
42 10
9ix 88
11.
98 18
3 x 96
12.
43 12
5|xl32
13.
22 10
Six 128
14.
3 15
6 X 176
!5.
10 11
8J X 270
16.
13 7
4jx275
8. When, however, the multiplier, though large, cannot be
broken up into factors, we must proceed as in the first case.
d. Here 3/. x37 = lll/ = 27rf. 3/, or 27|</.; we set
4| down frf., and carry 27d, :— 4rf. x 37 = 148rf., and, o^ld-
37 ing the 27dl, we have 175</. = 145. 7rf.; wc set down
Ex.
£
23
8,
11
£872 1 7| 7rf. and carry 145. ; and so on.
• In order to find these, note that any no. is exactly divisible by 5, if it ends in 5
or ; by 2, 4, 8, if the no. formed by it-j last one, two, three figs, respectively is div.
9, respectively,
and respectively ;
8, since 628 is by 8.
it should be used
thus42(f.xl2s:4|«.
by 2, 4, 8 ; by 3 or 9, if the sum of its figures is divisible by S or
Thus 76 and 80 are each divisible by 6, since they end in 6
84 by 2, since 4 is div. by 2 ; 766 by 4, since 56 is by 4 ; 1528 by
When in'the multiplication of £. s. d. one of the factors is 12,
first, as it raises the no. of the pence to the tame no. of shillings ;
B4f. 9d,
COMPOUKD DIVISION.
21
£ 8, d.
£ 8, d.
Ex. 7. 1. 43 8 6J X 19
2. 47 13 2jx 23
3. 33 15 Sjx 29
4. 79 16 3 X 34
5. 18 15 2jx 47
6. 24 14 31 X 62
7. 19 10 8ix 79
8. 15 17 42 X 93
9. 23 18 6^x106
10. 16 13 7|xl39
X
11. 3 qrs. 6 lbs. 13 oz. 15 dr. x
8
12. 4 tons 13cwt. 171b. lOoz.
9
13. 5 tons 27 cwt. 27 lb. 5 oz. x
25
14. 9 tons 16 cwt. 1 qr. 5 oz.
X
32
15. 17 cwt. 3 qrs. 15 oz. 7 dr. x
36
16. 18 tons 3 qrs. 5 lb. 13 drs.
X
45
17. 3 lbs. 8oz. 15dwts. 13grs. x
49
18. 2 lb. 7 oz. 9 dwts. 22 grs.
X
50
19. 5 fur. 78 yds. 2 ft. 7 in, x
56
20. 7 fur. 87 yds. 1ft. Sin.
X
04
21. 5a. 3r. 27p. X
70
22. 17a. IR. 3lP.
X
72
23. 3 sq. yds. 8 ft. 131 in. x
80
24. 17 cub. yds. 21 ft. 57 in.
X
84
25. 87 gals. 3 qts. 1 pt x
90
26. 37 gals. 2 qts. 1 pt.
X
96
27. 4 qrs. 6 bus. 2 pks. x
100
28. 3 qrs. 5 bus. 2 pks.
X
108
29. 5d. 17h. 39m. 20s. x
120
30. 17yrs. 110 d. 17 h. 57 s.
X
144
Compound Division.
9. BuLE. When the divisor id an abstract no., set it in a
loop to the left of the dividend, and divide the left-hand
term by it, setting the quotient under that term : if there be
any rem^, reduce it to the next lower den'^, adding in that
term (if any) of the div^, which is of this lower den°, and
divide the result by the div*^ : and so on.
£ 8, d, ^®^® ^^^ ^® 1*^^® *o divide £38 by 3,
Ex. 1. 3 ) 38 6 8^ whence we get £\2 with £2 over : now, as we
£12 15 6| cannot divide £2 by 3, we reduce it to 40*., and
adding in the term 6*. in the dividend, we have
now to divide 46s. by 3: — hence we get I5s. with 1*. over; and since
l5. = 12d, adding in the term Sd, in the dividend, we have now to divide
20d, by 3 : — hence we get 6rf. with 2d, over; and since 2c?. « 8/., we
have lastly to divide S/l + lf.yOr 9/, by 3, which gives us Sf, or |</.
"£ 8, d. Here the number ofpounds is exactly divisible
Ex.2. 8) 376 2 6 by 8 ; and since we cannot divide the term, 2*.,
£47 32 of the dividend by 8, we reduce it to pence, and
adding in the term 6rf., we have now to divide
sod, by 8 ; whence we get 3d with rem' 6(/. j and since GdL=*I^J« ^^
divide 24/. by 8, and thus have 3/. or fd.
22
COMPOUND DIVISION.
Ex.8.
£ s.
d.
£ s.
d.
1.
26 15
S|-^ 2
2.
12 14
8|-^ 3
3.
56 15
8 -:
r 4
4.
76 17
2i^- 5
6.
84 10
3 -,
r 6
6.
90 13
8|-i- 7
7.
75 7
6 -
'r 8
8.
87 16
8j-r 9
9.
91 14
41.
rlO
iO.
74 17
7j-i-ll
11.
57 13
-
rl2
12.
87 13
6 -^12
N.B. Division, primarily, as already stated (p. 7), is the
method of finding how often one quantity is contained in
another of the same kind^ i.e. how often one quantity must
be taken to make up another. Compound Division of this
nature will be found treated in art. 13. The result is then a
quotient properly so called. But by the kind of Division ex-
emplified above it is required to find that quantity which is
contained a given number of times in a given quantity ; and
the strictly accurate procedure in the first worked Ex., viz.
dividing £38 6s. %\d, by 3, would be as follows : The 3rd part
of £38 is £12 with £2 over, the 3rd part of 46^. is 15*. with
1*. over, the 3rd part of 20\d, is 6c?. with 2^d, over, the 3rd
part of 9 farthings is \d.
10. Division by 10, 100, 1000, &c. is usually performed
hj pointing 0^ one, two, three, &c. figures, respectively, from
the riffht of the dividend.
^ Here, dividing 2315 by 100, we have a quotient
7 23 with rem^ 15 ; we may point off, therefore, the
last two figures as the rem', leaving the rest for the
quotient ; reducing now this rem' into shillings,
and adding in the term 14*., we have to divide
314*. by 100 ; and since the quotient is 3 with
rem' 14, we may again point off the last tivo
Ex.
£ s.
23.15 14
20
3.14*.
12
1,75(1.
4
3.00/
figures as the
rem'
^: and
so on.
£ 8,
d.
£ 8, d.
Ex. 9. 1. 176 16
8-7-10
2.
30 6 3-rlO
3. 329 1
3-rlOO
4.
73 12 ll-^100
5. 1511 9
2-MOOO
6.
72 18 4-f 1000
7. 645 16
8 -i- 10000
8.
1062 10 0-MOOOO
11. When the divisor islarge,but can be broken up into two
or more factors, we may divide separately by each of these.
COMPOUND DIVISION.
23
£ s.
d.
£
s.
d.
Ex.10. I
702 6
3 ^ 20
2.
187
14
11 -
r 14
3.
275 15
l|-r 18
4.
345
13
4 -
r 40
5.
345 10
5 ^ 25
6.
351
14
8 -
r a2
7.
485 17
6 -^ 120
8.
457
18
4 -
r 400
9.
208 16
9 -r 36
10.
3*^2
19
101-
r 42
11.
692 10
-i- 800
12.
1137
10
-:
■-2400
13.
347 1
3 -^ 45
14.
457
1
n-
- 63
15.
362 10
-^6000
16.
1556
5
-i
-3600
17.
408
9 -i- 54
18.
453
11
ejH
- 77
19.
363 18
2i-^ 81
51 -^ 99
20.
473
14
-i
- 96
21.
386 16
22.
374
19
3 -J
- 108
23.
319 2
9 -r- 132
24.
576
3
-i
- 144
12. When, however, the divisor, tliough large, cannot be
broken up into factors, we must proceed as in the first case,
only setting the quotient in a loop at the right of the divi-
dend, instead of under it.
Ex. Divide £3715 18*. 9d. by 470.
Since 470=47 x 10, the sum may stand thus i
£ 8, d' £ 8. d
47)3715 18 9 (79 1 3
329
425
Here the rem' from the pounds is
£2, which we
423
]
reduce into shillings, adding
in the term IBs. in the
2
d.ividcnd: and
so on.
20
^We have now to divide this
I first quotient by 10 :
58(1
£ 8,
d.
47
10) 7.9 I
3
11
20
12
18.1
141(3
12
141
1.5
4
Am
'. £7 185. ll<f.
2.0
£
8.
d
£
8. d.
11. 1. 375
13
51-
1- 13
2. 289
81 f-
17
3. 258
1
8 -
^- 190
4. 456
lll-r
23
5. 371
2
9^
r 29
6. 513
8 9 -r
3100
7. 412
21.
'- 370
8. 712
18 7l-r
41
9. 1375
13
62-
1- 123
10. 2559
7 6 -r
18900
11. 2456
2
11 -.
-3650
12. 2348
11 41-r
• 354
13. Hitherto we have had to divide some quantity of
money, weight, &c., or, as it is called, some concrete o^'axvN-Vcjn
bj & simple, or abstract, number, that is lo ^tv^^^^V^^^^^^^^
24 COMPOUND DIYISIGN.
to find a certain part of such a quantity : thus, to divide
£3 7*. 6ff. by 8, is to find the eighth part of £3 7«. 6d. ; and
here the result will also be a concrete quantity of the same
hind as the dividend — ^as in this case, 8^. 6\d.
But if we have to divide a concrete quantity by another
of the same hind, this amounts to finding how many times
the divisor is contained in the dividend : 'thus, to divide
£3 7^. Qd, by 16*. \0\d,, is to find how many times 16*. I0\d.
is contained in £3 7*. Qd, ; and here the quotient will be an
abstract number — as in this case, 4.
The quotient in cases of this latter kind is to be found by
reducing the two quantities to the same denomination^ and
then performing the division.
Ex. 1. Divide £^ 7s. 6d. &yl6s. lOji
Here £3 7 6 =1620 halfpence \ hence 405) 1620 (4 Ana.
16 10i= 405 halfpence J 1620
Ex. 2. Divide 3 tons 2 cwt 1 qr. 21 lbs. by 2 qrs. 7 lbs.
tons cwt. qr. lbs. lbs.
Here 3 2 1 21 = 6993 \ hence 63) 6993 (1 1 1 Ans,
2 7= 63J 63^
69
63
Ex. 12.
£ 9.
1. 11 7
3. 102 10
5. 68 '6 lOi-f- 7 11 lOj
7. 684 7 6 -r76 10
9. 89 cwt. 22 lb. -r 3 cwt 1 qr. Gib.
11. 81 cwt Iqr. 16lb.-^lcwt. 3qr. 16 lb.
12. 9 1b. 9oz. 3dwts. 12 grs. -7-5(1 wts. 9grs.
13. 513m,4fur. 23po.-rl7m. 5 fur. 27 po.
14. 1027 m. 1 fur. 6 po.-r 17 m. 5 fur. 27 po.
15. 244 qrs. 3 bus. 1 pk. -7-3 qrs. 3 pks. 16. 2366a. 3r. 36p. +91a. 6p
14. To this head also may be referred certain cases of
Reduction, in which we cannot pass directly, step by step,
from one den** to another, but must reduce both the given
quantity and the proposed to some common lower den°, (it
nr/JJ be best to take the highest detf^ to ^Vv\Qi\i t\i^^ ^^\i\iQ\}cw
d. £
8, d.
914- 1
5 3J
35-^n
7 9?
63
63
£ s. d. £ 8. d.
2.
22 15 7j4- 3 15 111
4.
68 6 IQi- 2 10 7|
6.
205 7i-r34 3 5i
8.
171 1 10i4-57 7|
0.
1 95 m. 7 fur. -r 7ft. 6 in.
EECTASGDLAK AEHAS. 25
be reduced), and tLen find by dJv" what quantity of the pro-
posed dea' is equivalent to the given quantity,
Ex. Etduee £96 1 6s. to guineai.
£96 16s.
20 Here, since we know tliat 21j. make a guinea, we firrt
81) 1936 (92 TcducB the given sum into shillingB, and (ben divide
189 bj 21, to bring these ehillings into guineas. The rem'
46 4 vfo «t down (Art. 3., Ex, 3.) aa 4»,
42
B«i]ace Ex. 13.
1. 835 giiineaa to pounds ; and 538 pounds to halfguinea*.
2. 760 hiircroivns (o gntneas ; and 670 balfgiiineas to halferoirni.
3. 339 Clowns to balfguincas ; and 253 guineas to crowns.
4. 18756 fourpenny -pieces to crowns ; and 3700 halfcrowna to four-
penny- pieces.
5. £36 17s. 6J. to crowns J and £27 5$, 4d. to sixpences.
6. 100 lialfguineas to fourpcnny-pieces ■, and £100 to seven-
ehillmg-pieces.
7. t cwt. alba. Av. to Troy weight ; and 16 dwta. to Ap, weight.
8. 20 lbs. Av. to Troy weight ; and 5 drs. Ap. to Troy weight,
9. 478 ells to yards i and 14 hands to feet.
10. 500 fathoms to yards; and 5 farlonga to fathoms.
15. It must be noticed, that we can never divide a con-
crete quantity of one kind by another of a different kind, aa
shillings by ounce?, pounds by hours, &c. 5 since no quantity
of shillings will contain ounces, nor of pounds, hours, &c.
Nor can we multiply togi,t!ier concrete quantities of ant/
kind, whether the same or different : thu», we cannot mul-
liply either shillings by shillings, or shillings by ounces.
16. Mensuration of rectangular areas.
^ Suppose ABCD to represent the surface of
a table, of which the length AB is 5 feet, and
the breadth AD, 3 feet. Divide then AB into
5 equal parts, and AD into 3, aa in the figure,
ind through the points of divisioTi it^w \iQ«^
-^ parallel to AB and AD. "Bj ftvs ■a».wa& -«*
26 RECTANGULAR AREAS.
shall have divided the whole surface into small figures, such
as AEFG, all equal to one another ; and since AE =^oi\q
foot, and AG=^ one foot, it is plain that the sur^acQ AEFG
measures a foot every way, a foot long and a foot broad, — i. e.
AEFG is a square footy and so are all the other small figures.
Now the number of these figures is 5x3= 15, each
horizontal row of 3 square feet (the number of feet in AD)
being repeated 5 times (the number of feet in AB) ; so that
the number of square feet in the surface is found by multi-
plying together the n® of feet in its length and the nP of
feet in its breadth.
17. As the same method of proof would apply in any
similar case, it appears that the d9 of square feet in any
rectangular surface is found by multiplying together the n°
of linear feet in its length and breadth; or if we express
the length and breadth in yards, inches, &c., and multiply
them in this form, we shall obtain the n® of square yards^
square inches^ &c. in the surface.
Ex. Find the surface of a floor 17 ft. 8 in, long by 3 yards wide.
Here 17 fit. 8 in. = 212 in.* 12)_22896
3 yds. = 108 in. 12 ) 1908
1696 9) 159
2120 — fTHs
22896 sq. in. = 17 sq. yds. 6 ft. Ans,
Ex. 1ft. 1. 37 ft. 2 in. X 2 ft. 9 in. 2. 23 ft. x 3 ft. 5 in.
3. 3 yds. 2 in. X 3 ft. 4. 1 yd. 2 ft. x 1 yd. 1 in.
5. 15 ft 7 in. X 11 ft. 11 in. 6. 22 ft. 5 in. x 3 yds.
7. What is the area of a court, 10 yds. 2 ft;, long, and 5 yds. 1 ft.
broad ?
8. How many sq. yds. of carpet will it take for a room 26 ft. by 32 ft.?
9. What is the surface of a marble slab, whose length is 5 ft. 7 in., and
breadth 1 ft. 10 in.?
10. Find the area of a square building, whose side is 46 ft. 8 in.
* It might at first (tight appear that we are here multiplying inchet by incheSy con.
trary to the statement In (15) ; but, in reality, it is only the numbers 212 and lOd that
we multiply, not the quantities 212 in. and 108 in. : so also the resulting product is
only the number 22S96, to which we append sq. in., because we know from tne above,
that this is the number of square inches in the given area. A similar remark applies
to all such cases, and to all such expressions m multiplying the length by the brcadtk,
&c. The Student's attention should be strongly drawn to this.
RECTANGULAR AREAS, 27
11. How many square yards of paper will be required for a room 17 ft.
long, 12 ft. 7 in. wide, and 8 ft. 5 in. high ?
12. How much wainscoting is there in a square room, 18 ft. 3 in. long,
and 8 ft. 6 in. high ? See Note IV.
18. Since, by multiplying the length and breadth, we get
the square area of any rectangular surface, it follows that,
by dividing the square area by the lengthy we shall get the
breadthy or, dividing it by the breadth, we shall get the
length — taking care to express the quantities concerned,
before div**, as quantities of the same den°, as, for instance,
not dividing sq. feet by inches, but first bringing them to
sq. inches, &c.
Ex. What length of wall paper, 2 ft. wide, will 6e required for a
room 14 ft. square, and 10 ft. 4 in. high ?
The room being square, the united length of its four sides will be
14x4 = 56 feet, and their height being 10 ft. 4 in., we shall find the
square area of the whole surface of the walls by multiplying these
quantities, first reducing them to inches.
Here 56 ft. = 672 in. The surface of the walls being 83328
10 ft. 4 in. s =124 in. gq. in., we have now to divide this by
2688 2 ft. =24 in., the width of the paper.
1344 in. iq. in. in.
672 24) 83328 (3472
72
83328 sq. in.
113
96
Am, 3472 in. =96 yds. 1 ft;. 4 in. 172
168
48
48
Ex.18.
1. 5 sq. yds. 6 ft. 15 in. ^- 18 ft. 7 in. 2. llsq.yds.3ft.129in-i.2ft.9in.
3. 8 sq. yds. 6 ft. 84 in.-^5 ft. 9 in. 4. 17 sq. yds. 4 ft. 24 in. -5-23 ft.
5. 17 sq. yds. ft. 45 in. -5- 18 yds. 1 ft. 9 in.
6. 42 sq. yds. 1 ft. 50 in.-^23 ft. 10 in.
7. Find the length of a room, 1 1 ft. 11 in. wide, the Hoor of which
it takes 17 sq. yds. 2 ft. 131 in. of drugget to cover.
8. One side of a rectangular building measures 26 yds. 5 in., and its
area contains 683 sq. yds. 2 ft. 25 in. ; show that it is square.
9. How many yards of carpeting, 2 ft. 4 in. broad, will it take to cover
a room whose dimensions are 26 ft. by 35 ft. ?
1 0. It is found that 288 yds. of paper, 2 ft. 8 in. wide, will coN«t >53Qa'«^^BSs&
of a room ; how many would be required oi po^pex 'H\.,^\EL%Vi.^^t
28 . EECTANGULAB SOLIDS.
11. How many yards of matting, 2 ft. 3 in. wide, will be required for a
square room, whose side is 18 fb. in.?
12. If the room in (11) be 13 ft. 4 in. high, how many yards of paper
1 ft. 4 in. wide will be required for its walls ?
19. Mensuration of rectangular solids.
Suppose we place upon each of the little squares in the
preceding figure, a solid (as, for instance, a brick) in the
form of a ciibic foot, that is, measuring a foot every waj —
a foot long, a foot broad and a foot high — we shall have a
layer of such bricks one foot high, and containing as many
cubic feet as there are square feet in the base ; if upon this
we pile another similar layer, we shall have the whole solid
two feet high, and containing twice as many cubic feet as
there are square feet in the base ; and so on ; hence the
whole n® of cubic feet in any such solid, will be found by
taking the product of the n^ of feet in height by the n** of
square feet in the base, and this last, as in (17), is the pro-
duct of the n^ of feet in length by the ia9 of feet in breadth.
Hence the n® of cubic feet in any rectangular solid or
space is found by multiplying together its lengthy breadth,
and height (or thichness, as the height would be called when
small, as, for instance, in the case of a beam of timber), these
quantities being all reduced first to the same den**, and their
product being of the same den°, but in cubic measure.
Ex. Find the solid content of a beam of timber, 30 ft. long, 2 ft. 3 in.
wide, and 2 ft. 5 in. thick.
in.
Here 30 ft.«. 360 1728) 281880 (163 cub. ft;.
2 ft. 3 in. =r 27 1728
2520
720
10908
10368
27) 163 (6 cub. yds.
162
9720 sq. in.
2ft.5in.= 29
..5400
5184
1ft.
87480
19440
216 cub.
in.
Ans. 281880 cub. in. =6 cub. yds. 1 ft. 216 in. by Red".
MISCELLANEOUS EXAMPLES. 29
20. So also, as before, having given the cubic content of
any space and any two of its three dimensions, we may find
the third by dividing the content by the product of these
two, reducing all to the same den".
Ex. What is the length of a room, whose width is 10 ft. 4 in. and
height 10 ft. 6 in. ; and which contains 1519 cub. fl. of air ?
and 1519 cub. ft. =2624832 cub. in.,
h' nee, performing the div», we have
sq. in. cub. in. in.
15624) 2624832 (168
15624
10G243
93744
Here
10 ft.
4 in, = 124
in.
10 ft.
6in.»l26
744
1488
ia>
1.5624
sq. in.
Ans.
168 in.
= 14 ft.
1*24992
124992
Ex.l«.
1. 18ft. 9in. X 13ft. 4 in. x 8 ft. 4in. 2. 3ft. 9in. x 6ft. 8 in. x 2ft.l in.
3. Ilft.3in.x3ft. 4in. xlOft.5in.
5. 7 ft. 4 in. x 5 yds. x 8 ft. 3 in.
4. 5 yds. X 6 yds. 2 ft. X 4ft. 2 in.
6. 9 ft. 2 in. X 2 yds. x 6 ft. 8 in.
7. How many cubic feet of water can be contained in a vessel with
square base, whose side is 3 ft. and height 2 ft. 10 in. ?
8. What quantity of timber is there in a beam, whose length is 20
feet, breadth 3 feet, and thickness 2 ft. 6 in. ?
9. Find the solid content of a cube, whose side is 7 ft. 5 in.
10. In making a square pond, whose side was 1 2 yds., there were
taken out 336 cub. yards of earth ; how deep was it made ?
11. What must be the length of a trench, 6 ft. 6 in. deep, and 10 ft.
8 in. wide, that it may contain 7040 cubic feet ?
12. The depth of a canal is 7 ft. 3 in., the width 20 ft. 4 in., and the
length 10 miles ; how many cubic feet of water will it con-
tain ?
Miscellaneous Examples. 17.
1. A sovereign weighs nearly 493 quarter grains; how many lbs. will
1000 sovereigns weigh?
2. In 2651443 seconds, which is the exact length of the lunar
month, how many days?
3. What is the cost of 530 lbs. of tea at 3s. *ld. per lb ?
4. Six persons on a journey spend £97 95. 6c?. ; how much is that
for each person ?
5. The circumference of the Earth contains 1 31260000 feet \ «r^^«e&
the same in milei|.
P
so HISCELLANBOUS EXUfPLBS
6. If 81 oxen are bought for £1779 19«. 6i., what is the average
price per head ?
7. How many letters, paying penny postage, require stamps to the
amount of £7947 28, lOd, ?
8. A pint will contain 9000 barleycorns, and 3 of these, placed end
to end, would reach an inch ; how many feet would they all reach ?
9. How many days would it take to count a million of soyereigns, at
the rate of 100 a minute?
10. What is the amount of 42 cwt. of sugar at £2 35. 7d, per cwt. ?
11. Divide 3587 yds. 9 in. into 27 equal distances.
12. What sum must be divided among 27 men, so that each may
receive £14 6s. 8^. ?
13. How many ducats, each worth is. 9d., are contained in £231 165. ?
14. Divide £1478 12*. 9^. into 77 equal portions.
15. How many days in a solar year, which contains 31556928
seconds ?
16. A cubic foot of water weighs 1000 ounces ; what weight of water
is there in a vessel, the length, width, and depth of which are each a yard ?
17. The battering ram employed by Titus against the walls of Jeru-
salem weighed 100000 lbs. ; how many tons did it contain ?
18. The Calcutta rupee is worth Is. l}^,; what is the value of a
lac, which consists of 100000 rupees?
19. Sound travels at the rate of 1 130 feet a second ; how many miles
is a thunder-cloud distant, when the sound follows the flash after 7
seconds ?
20. Light travels at the rate of 186010 miles a second ; if the Sun*s
light takes 8 min. 13 sec. in reaching us, what is his distance from the
Earth?
21. A cannon-ball travels at the rate of 400 yards a second ; how
many miles will it go in a quarter of a minute ?
22. Find the amount of 200 tons 81 lbs. of iron railing at 7d. per lb.
23. Suppose a weekly newspaper, price 3(?., has a circulation of
11800 ; what is the sum realised by its sale in a year ?
24. If 2 cvt. I lb. cost £116 Ids. 0^., what is the cost of 1 lb. ?
25. How much silk at 6^. Sd. a yard may be bought for 20 guineas ?
26. To how many persons may £60 155. Od. be distributed, giving
£4 135. 6<^. to each?
27. An Attic drachma was worth 7^. ; what was the value of the
talent, which contained 6000 drachmae? and how many minse did it
contain, each worth £3 4*. 7d. ?
28. A Jewish shekel weighed 219 troy grains, and was worth 25.
3^. ; what was the weight of a talent, containing 3000 shekels ? and
the value of 10000 talents?
IK ELEMENTABT BULES. 81
29. The captains of Israel, after the destruction Of Midian, made a
free-will of^ng of 16750 shekels ; what sum did this amount to ? See
Ex,2S.
30. How long would a cannon-ball, moving at the rate of 1200 foet
ft second, be in passing from the Earth to the Moon, 237600 miles ?
31. How much is spent in 15 years bj a person who spends :£825
185. 9d. yearly? and how much would he have saved in that time out
of an income of £1500 ?
32. How many pounds weight of bronze are there in a million of
pennies, each weighing one-third of an ounce avoird. ?
33. A plate of gold cost £161 17«. 6d,, at £4 7^. 6<f. per ounce;
what was its weight ?
34. How many patients will an hospital maintain, whose revenue is
£5629 10^., when each requires on an average £8 13«. 9d. per annum ?
35. If the duty on brandy, at 10». 6d, a gallon, amounted to £26357
be, lOd., on what quantity was it paid ?
36. Twenty bricklayers and ten carpenters were employed in build-
ing a house, each of the former receiving 27«. per week, and each of the
latter 298. ; what was the amount of their wages in 16 weeks ?
37. Two boats start in a race, and one of them gains 5 feet upon the
other in every 55 yards ; how much will it have gained at the end of
half a mile?
38. What is the area of a playground 58 ft. 6 in. long, and 54 ft
9 in. broad ?
39. A has £100 4«. 1 1^., and B 64393 farthings ; if A receive from
^11111 farthings, and B from ^ £11 Us, 11^., how much will ^ have
more than B ?
40. What is the value of a beam of timber, whose length is 20 ft.,
breadth 3 ft, and thickness 2 ft., at 3». SJef. per cubic foot?
41. If the length of a cubit was 22 inches, what was the cubic
content of the Ark, which was 300 cubits long, 60 broad, and 30 high ?
42. A grocer mixes 3 cwt. 24 lbs. of sugar at 6^. per lb. with 2 cwt.
64 lbs. at 4J<?. ; at what price per lb. must he sell the mixture, so as not
to lose by the sale ?
43. A person gives a flvo-pound note to pay for lodgings during the
month of August, at 25. Sd. per night ; what sum will be returned to
him?
44. Of the three quantities 1347 lbs. avoird., 449 shillings, and
£6286, it is required to multiply one quantity by the quotient of the
other two.
45. What is the cost of 6 packs of cloth, each containing 6 parcels,
each parcel 6 pieces, and each piece 60 yards, at 2^. per yard?
46. A labourer's house-rent is £6 28. l\d, a yea-T •, "^\^\> m^^&\.V^^a.'^
op weekly to pay it ?
D %
S2 MISCELLANEOUS EXAMPLES
47. It is estimated that the average strength of a man is equal to
raising 100 lbs. through 1 foot in a second, working 10 hours a day;
how many tons will he raise at this rate in the day ?
48. In marching, soldiers take 75 steps a minute, in quick marching
108 ; how far would a regiment advance in 3 hours, the last half-hour
at quick march, reckoning each step as 2 ft. 8 in. ?
49. If a compositor set up 8500 letters a day, and be paid 6^, for
every thousand, how much will he earn in a week ?
50. Divide £184 11^. 2|^. equally among 39 persons; and, sup-
posing 15 of them to have received their portions, and of the rest only
21 to appear, how much might be given to each of these?
51. A mixture is made of 9 gallons of spirits at 12«. 6d, per gal.,
16 gallons at ISs. 9d., and 90 gallons at 22^. dd. ; what is the value of
a gallon of it ?
52. A com-£9ictor buys 2 quarters at 39s. per quarter, and 7 bushels
at 6«. per bushel ; at what price per bushel must the whole be sold, so
as to gain 238. 9d. in all ?
53. A side of Lincoln's Inn Square is 770 feet, and of Bussell Square
670 feet ; how many acres does each contain ?
54. What weight of water may be contained in a canal whose depth
is 8 feet, width 25 feet, and length 12 miles? See Ex. 16.
55. How many yards of carpet, 25 inches wide, will be required to
cover a floor that is 19 ft. 7 in. long by 18 ft. 9 in. wide?
56. A wished to exchange 50 gallons of brandy, at 2l8. 9d. per gal-
lon, with By for ale at Is. 6d. per gallon ; how many gallons of ale
should he receive?
57. A wall is to be built, 15 yards long, 7 feet high, and 13 inches
thick, with a doorway 6 feet high and 4 feet wide ; how many bricks
will it require, if each, including mortar, occupy 108 cubic inches?
58. Divide £115 10s. among 5 men and 6 women, giving to each
man thrice as much as to a woman.
59. An equal number of men, women, and boys earned £55 13*. in
6 weeks ; each man earned 2s. id. a day, each woman Is. dd.j and each
boy lOd. ; how many were there of each ?
60. There is a plantation in the form of a hollow square, length ex-
ternally 252 yards, and depth 16 yards ; find the area of the plantation
and that of the inner square.
61. Divide £39 into four equal numbers of guineas, half-guineas,
crowns, and half-crowns respectively.
62. A clergyman commutes his tithes, valued at £500, for equal
quantities of wheat, barley, and oats ; how much grain will he receive,
supposing the average price of wheat to be 6s. 7d, a bushel, of barley
3s. lld.f and of oats 2s. lOd. ?
or ELEHENTABT BULBS.
83
63. A and B go to bed at the same hour daily, but A rises at a
quarter past 6, and B &tS; how much of waking life will A have had
more than B in 40 years, paying attention to the Leap-years ?
64. Divide £20 among tiiree persons, so that one may have £3 15jl
more than each of the others.
65. Divide £550 3^. 1^. among 4 men, 6 women, and 8 children,
giving to each man double of a woman, and to each woman triple of a
child.
66. Divide £2 9«. 2d, among A, B, C, bo that B may have Gs, Sd.
more than A, and Cb share may be double of ^s.
67. The circumference of the fore wheel of a carriage being 8 ft.
3 in., and that of the hind wheel 11 ft. 11 in., how many more revolu-
tions would be made by the fore wheel than by the hind wheel in going
from Cambridge to London, a distance of 52 miles ?
68. In each of the two subjoined tables, add sideways the nos. in
line a, placing the total at the end of the line. Do the same with each
succeeding line. Then add upwards each of the columns A, B, &c.,
placing its amount below it. Lastly, find the gross total both by total
line and total column, to verify the result.
(i)
A
B
C
D
E
F
Total
a
2037
586
1913
4960
881
2476
b
2666
1389
809
1547
1529
3008
c
745
1748
676
2731
1164
903
d
3567
2607
1300
800
1920
966
e
494
464
2796
3794
768
1760
/
1892
3025
815
839
808
2913
9
1439
1263
1557
1106
3650
972
Total
(ii)
A
B
C
D
E
F
Total
a
629
570
561
1053
1297
1008
b
2785
2259
960
3786
3576
730
c
856
850
3757
4084
914
2109
d
1793
3725
849
1691
1930
2364
e
3659
4671
1472
778
2081
4060
f
3087
1968
1536
2357
617
960
9
944
1706
2817
1809
4507
1677
Total
^
'v
\
84
CHAPTER IL
GREATEST COMMON MEASURE: LEAST COMMON MULTIPLE.
21. One number is said to be a measure or 9k factor of
another, when it divides it exactly, without remainder.
Thus, 1, 2, 3, 4, 6, 12 are all measares or factors of 12.
Unity^ however, is not generally named among the divisors
of a number.
22. Anj number, which divides without remainder each
of two or more numbers, is said to be a common measure or
common factor of those numbers ; and, of course, the greatest
number which so divides them is their Greatest Common
Measure (g. c. m.)
Thas 2 is the only common measure of 4 and 6; 3, 5, 15 are, each of
them, common measures of 30 and 45, and 15 is their greatest common
measure; 2, 7, 14 are, each of them, common measures of 14, 42, and
70, and 14 is their greatest common measure.
23. To find the Greatest Common Measure of two numbers.
Rule. Divide the greater by the less, and the preceding
divisor by the remainder, and so on continually, until there
is no remainder : the last divisor will be the g. c. m. re-
quired.
Ex. 1. Find the a. c. k. of 3575 and 125455; and of 279 and 4185.
8575) 125455 (35 279) 4185^15
10725 279
18205 1395
17875 1395
830) 3575 (ID Ans. 279.
330
275) 330 (1
275
55) 275 (5
275
Ans. 55.
35
LEAST COMMON MULTIPLE.
Ex. 2. find the o. c. x. of 17 and 36.
17) 36 (2
34
2)17(8
I?
1)2(2
2
Ans, 1.; L e. the given numbers have no common measure but uni/y.
The reason of this Rule can hardly be explained without some know-
ledge of Algebra in the Student The Rule itself is here introducedi
because it is often useful in reducing Vulgar Fractions to simple forms.
See NoTB V,
Kx.18.
■ Find the G. c. k. of
1.
224 and 336.
2.
348 and 1024.
3.
175 and 2042.
4.
1225 and 625.
5.
2121 and 1313.
6.
429 and 715.
7.
377 and 1131.
8.
2431 and 770.
9.
900 and 3474.
10.
1379 and 2401.
11.
2314 and 3721.
12.
7007 and 7392.
13.
2793 and 2660.
14.
4165 and 686.
15.
5325 and 8307.
16.
3775 and 10000
17.
7056 and 7392.
18.
6327 and 23997.
19.
12321 and 54345.
20.
24720 and 4155.
24. One number is said to contain^ or to be a multiple of,
another, when it can be divided bj it without remainder.
Thus 12 is a multiple of each of 1, 2, 3, 4, 6, 12 ; and any number is
a multiple of each of its measures.
25. A common multiple of two or more numbers is one
which contains each of them ; and, of course, the least such
number, is their Least Common Multiple (l. o. m.).
Thus 6, 12, 18, &c., are all common multiples of 2 and 3 ; but 6 is
their least common multiple i 12, 24, 36, 48, &c., are all common multi-
ples of 2, 3, 4, 6, and 12; but 12 is their least common multiple.
Of course, a common multiple of any given numbers may
be found, by multiplying them all together ; thus a common
multiple of 6 and 8 is 48, of 4, 6, and 9 is 216. In practices,
however, we require the least common multiple.^ e%^^w!^
in preparing Vulgar Fractions for Ad4\l\oiv wv^^xOci^x^^'^'^^'^'
3G LSAST COMMON MTTLTIPLB.
26. To find the Least Common Multiple of two or more
numbers.
EuLE. Set them in a line, and strike out any that are
contained in any of the others. Divide those not struck out
by any number that will exactly divide one of them; under
any which it exactly measures, place the corresponding quo-
tient ; under any which it partially measures (containing
some factor common to it, but not being itself wholly con-
tained in it), place the quotient obtained by dividing it by
the common factor; and under any which it does not mea-
sure at all, repeat the number itself.
Now treat the new line thus formed, in the same manner
as the first; and so on, until all the numbers left in any line
have no common measure but unity.
Then the continued product of the numbers in this line
and all the divisors is the L. c. m. required of the given
numbers.
Obs. It will generally be most convenient to take pretty
large numbers, if possible, for divisors; as fewer lines will
thus be necessary, especially if such be chosen as contain
themselves many simple factors. Thus 12 contains the fac-
tors 2, 3, 4, 6, 12, and is therefore, when possible, a very
good divisor to be employed.
Ex. 1. rind the l. c. m. of 24, 16, 6, 20, 4, 8, 10, 30, 12, 25.
12 ) 24 . 16 . y? 20 . ^ . )^ . X^? > 30 . t,!^ . 25
^ . 4 . 1^ . \ . 25
Ans. 4x25x12 = 1200.
The reason of this process may be thus explained.
We are required to find a number, which shall contain 24, 16, 6, 20, 4
8, 10, 30, 12, and 25. Now if we find a number which contains 24, it
will, of course, contain 6, 4, 8, and 12, which are themselves contained
in 24. We may therefore strike out 6, 4, 8, and 12; and for a similar
reason, 10, which is contained in 20; and we thus reduce the question to
finding the l. g. m. of 24, 16, 20, 30, and 25.
Now "we choose for divisor, according to the Rule, the number 1 2,
which exactly divides one of these, viz. 24. In order, therefore, that the
L. c. u. required may contain 24, it must, of course, contain this number
LEAST COMMON MULTIPLB. SI
12, and besides that a factor 2 ; but we now wish to find what factors
basides 12 and 2, the l. c. m. must contain, so as to contain all the given
numbers. We see then that 12 will also supply the factors 4 of 16, 4 of
20, and 6 of 30 ; so that the only others besides 12, which must be con-
tained in the required number, are 2 to make up 24, 4 for 16, 5 fur 20,
5 for 30, and 25, ie. the numbers given by our process in the second liiic{
— to which a similar reasoning applies.
Ex 2. What is the least number that can be divided by each if K-
16,40,50,25,8,64?
10 ) 14 . 't,^ . 40 . 50 . H^^ . )S( . 64
7 . ^ . 5 . 32
^ns. 7x5x32 xlO» 11200.
£x. 3. Find the l. c. m. of 27, 24, 6, 15, 5, 9, 126.
9) 27.24.)^. 15 .^.^. 126
2 ) 3 . 8 . 5 . 14
3.4. 5. 7
Ans, 3x4x5x7x9x 2 = 7560.
Ex. 19. Find the i«i c. u. of
1. 15,20. 2. 14,21.
3. 8, 4, 16. 4. 3, 9, 22.
5. 12,15,16. 6. 8,16,20.
7. 9, 15, 18, 20. 8. 16, 9, 12, 18.
9. 8, 12, 15, 20. 10. 34, 68, 17, 2.
11. 6, 12, 16, 18, 24. 12. 8, 12, 18, 24, 27.
13. 2,4,8,16,10,48. 14. 1.2,3,4,5,6,7,8,9.
15. 7, 12, 15, 27, 35, 40, 45. 16. 9, 1 6, 42, 63, 21, 14, 72.
17. 4, 9, 10, 15, 18, 20, 21. 18. 7, 15, 21, 28, 35, 100, 125.
19. 8, 9, 10, 12, 25, 32, 75, 80. 23. 15, 16, 18, 20, 24, 25, 27, 3a
88
CHAPTEB m.
VULGAE FRACTIONS,
27. A Fraction is a quantity which represents a part or
parts of an integer, or whole.
28. A Vulgar (that is, a common) Fraction, in its sim-
plest form, is expressed by means of two numbers placed one
over the other, with a line between them.
The lower of these is called the Denominator^ and shows
into how many equal parts the whole is divided ; the upper
is called the Numerator, and shows how many of those parts
are taken to form the fraction.
Thus I denotes that the whole is divided into four equal parts, and
that three of them are taken to form the fraction.
29. A proper fraction is one whose numerator is less than
the denominator, and which is itself therefore less than the
whole in question ; as |, f .
An improper fraction is one, whose numerator is equal to
or greater than the denominator, and which is itself, there-
fore, equal to or greater than the whole in question ; as f , A^-.
30. A mixed number is one formed of a whole number
and a fraction ; as 2f , 5|.
A compound fraction is a fraction of a fraction ; as f of f ,
21 of f of 3^.
A complex fraction is one in which either the num', or
gl 2 li - of 3
den^ or both are fractions ; as ^» -pi -^, ^gT"*
YULGAB FRACTIONS. 39
31. Every whole number may be considered as a fraction
whose den*" is 1 ; thus 6 is f .
32. A fraction may be considered as expressing the divi-
sion of the num*' by the den'.
Thus I expresses 3•^4: for we should obtain the same, whether we
divide one unit into 4 equal parts, and then take three of these parts, that
is, three-fourths of the one unit; or divide three units, each into 4 equal
parts, and then take one part out of each four, t. e. one-fourth of each
unit, and therefore one-fourth of the whole three units; so that f of 1, or
|, = Jof 3, or 3+4.
For instance, | of j£l, which is 15&, =| o{ £% which is also IBs,
33. To reduce a whole number to a fraction with a given
denominator.
Rule. Multiply the number by the given den', and the
result will be the num^ of the fraction required.
Ex. Reduce 5 to a fraction, with denominator 6.
Since 1 contains 6 sixth parts, .\ 5 contains 30 sixth parts; or 5--^.
Ex. 20. 1. Reduce 8 and 27 to fractions with den" 5 and 27.
2. lieduce 34 and 135 to fractions with den" 11 and 17.
3. Reduce 6, 9, 12, 20, to fractions with den' 15.
4. Reduce 25, 34, 70, 111, to fractions with den' 34.
34. To reduce a mixed number to an improper fraction.
Rule. Multiply the whole number by the den' of the
fractional part ; add the result to the num' of that part for
the new num"", and retain the same den'.
Ex. 1. 7i-^: for 7=^ (33); and hence, 7|=-^ + |=^.
Ex.2. 1|-|. Ex.3. 6J=-^.
Ex. 21. Reduce to improper fractions
1. 3f, 2. lOf. 3. 221^. 4. 13if. 5. 32|^.
6. 200H. 7. 71i|. 8. 11512. 9. 128^. 10. 37lf.
11. 200|§. 12. 125||. 13. 514^. 14. lOll?. 15. 71911.
16. im- 17. 17iS|. 18. 10|1§. 19, lU^^^. ^IQ. '^^-^^^
40
YULGAB FBACTIONS.
35. To reduce an improper fraction to a whole or mixed
number.
EuLE. Divide the num' bj the den' : the quotient will be
a whole number^ and the remainder, if any, the num' of the
fractional part of the mixed number required.
Ex.1. ^=5. Ex.2. -^=7^
Obs. All improper fractions, occurring in anj sum, should
(except the contrary be desired) be expressed as whole or
mixed numbers.
Ex. ZZm Reduce to whole or mixed numbers
1 37 o 79 q 313 A
6 3127 7 1210 Q 2221 Q
• ■ 4§-. # • 66 • *'• " eV • '•
lO 8577 iq 4148 \A
•|7 5434 Ift 6556 IQ
11. ?^.
16. -^^.
2990
23 •
1247
4641
221 '
12321
200 *
5.
10.
15.
20.
1023
35 *
8136
8139
122 *
23438
~33S~*
36. To multiply a fraction by any whole number or integer,
either multiply the numerator, or divide the denominator
by it.
Ex.1. Ax7=it.
For in each of the fractions, ^ and ^, the whole is divided into 15
equal parts, and 7 times as many of them are taken in the latter case as
in the former.
Ex.2. ^x4=J=lf.
For the whole being divided into 4 times as many equal parts in ^ as
it is in |, each of the parts in the latter is 4 times as great as in the
former; and the same number of parts being taken in both cases, the
latter fraction is therefore 4 times as great as the former.
Ex.3. ix9=-¥-=5f. Ex.4. ^x4=f| = lif.
Ex.5. i?x9=-^=4l. Ex.6. Mx7=-¥-=3|*
37. Conversely — To divide a fraction by any integer, either
divide the numerator, or multiply the denominator by it.
Ex. 1. i|4-6«A-
Ex.3. 1^5=^
Ex.2.
Ex. 4.
28 • " 28*
9 • " — 6««
VULGAB FRACTIONS.
41
Ex. 23. 1. Multiply || by 9, 12, 18, 25 ; and divide it by 5, 7, 8, 12.
2. Multiply iff by 7, 8, 9, 16 ; and divide it by 5, 8, 12, 25.
a Multiply m by the numbers 2, 3, 4, 5, 7.
4. Divide f|§ by the numbers 7, 8, 9, 10, 11.
38. If the num' and den' of a fraction be both multiplied
or both divided hj the same number^ its value will not be
altered.
For if the nuwT be multiplied by any nimiber, the fraction is multiplied
by it (36), and if the denT be multiplied by any number, the fraction is
divided by it (37) ; and if any quantity be hoik multiplied and divided
by the same number, its value is not altered.
Similarly, when the nnm' or den' are both divided by the same
number.
39. To reduce a fraction to lower terms.
Rule. Divide both the num' and den' by any common
factors they contain.
Ex.l. »>|?f=»>fj-|f- Ex.2. »>iJf=^>||=|.
From (38) it appears that the value of a fraction is not altered by this
process.
When a fraction is reduced as much as possible by such
division, it is said to be in lis lowest terms. (See p. 20, note,)
Obs. All fractions, occurring in any sum, should (except
where the contrary is desired) be expressed in their lowest
terms.
Ex. 2ft. Beduce to their lowest terms
1.
6.
IL
16.
R24
aiTi'
3000
537f
6930
8118*
2.
7.
12.
17.
720
49S
T2T5'
2592
3458*
SS44
6552*
3.
8.
13.
18.
S24
SSI*
1296
1751*
1485
fl65*
7040
7555*
4.
9.
14.
19.
1584
F540*
1872
2016* *
864
11385
T553F*
5.
10.
15.
1290,
Tg2^
890
T§55*
3300
425?*
Oft 22176
40. A fraction may be reduced immediately to its lo^^^\.
terms by dividing both its num' and detf \>y ftiOT o. c.^su
42 VULGAR FRACTIONS.
This process is generally longer than the other, and is
therefore, if possible, avoided in practice. It is, however,
sometimes, the only way of reducing a fraction, when we are
unable to detect by inspection the conunon factors of the
num' and den^ Thus we should not see, perhaps, that the
fraction Jff ^ may be reduced to |f by dividing both its
terms by 113, their g. c. m.
Ex.1. "»>|iif-fi- Ex, 2. »«*>}?|if=it.
Ex. 25. Redace to their lowest terms
1 221 9 BIO a £SS d 140T
t 1905 fi 1715 7 6509 o 1589
9. mh' 10. m- 11. wh' 12. ms-
41. We shall now give examples of the application of the
foregoing rules to the mulf^ and div'' of concrete quantities.
£ 8. d.
Ex. 1. 23 13 9| X 35 jj^^e | x 5 =-^=4| ; we set down fd, and
carry 4d, :
so also I X 7 — -V-=2| ; we set down |<2., and
cany 2d,
Here, in the first div*, there are 5d.
Ex.2. 7)3';u ;-28 T"' '"^ ^^ ^''^'i^L^^^'^'^s'^l!^'
down as fdL, since (32) J of 5d.=f of Id.
3 We might have brought these 5d, to 20/
^ and then, dividing by 7, should have had
2f/ ; but as a farthing itself is only & fraction of a penny, it is usual,
when the result does not come out a clear number of farthings, to ex-
press the whole below the pence as a fraction of a penny.
In the second div», there is lf</. over, or -^., which, divided by 4,
gives f<f.
- , Here, in the first div", Uiere is Ijrf.
Ex. 3. 8) 175 1% .51-40 over^Jrf., which, divided by 8, gives
5) 21 19 11^ . ^- ; In the second div», there is 4^
— 2 — ly ,|f? over «= -^-</., which, divided by 5, gives
4 7 n„ ^
118 9
13
7
829 3
£
7) 37
91
8, d,
14 8-:
4) 6
7 9f
VULGAB FBACTI0N8.
43
£ 8. d. £ 9, d,
Ex. 4. 13) 54 10 5J (4 3 10||
52
2
20
50 (3«.
39
11
12
Here there is J^d. over^^-d, which, dWided
by 13, gives J|d
137|(10(/.
130
n
£ s, d.
Ex. 5. 3) 1115 17 8|. ^800
100} 3.71 19 2| H(§re there is 70|<f. over»i|^., which,
^20 divided by 100, gives J§8</. =H<'-
14.39
12
4.70| Ana. £3 14c. 4}2(f.
£8. d,
1. 3 17 4f X 5.
4. 7 8 llf xll.
7. 6 17 4f x32.
10. 5 3 4i|x31.
Ex. 26.
£ 8. d.
5 11 2J
2. 5 11 2J X 7.
5. 6 1 7^ X 15.
8. 2 19 9| x44.
n. 7 14 9£ x37.
3. 4 5| X 9.
6. 8 2 5| X 27.
9. 4 13 0{ X 29.
12. 6 18 Oj{^x41.
£ s.
£ 8,
d.
13.
2
1 H
-3.
14.
9 7
3i-r4.
15.
29 17
8 -
1-5.
16.
72 13
5 -i-6.
17.
8 13
H
-9.
18.
37 6
2 -^1Q
19.
73
^4 '
-8.
20.
29 7
01-7.
21.
69 17
-9.
22.
53 4
0i-rl2.
23.
124 15
6 -
rl5.
24.
131 11
8i-i-18.
25.
135 14
10 -
'rAO.
26.
111 11
lli-^60.
27.
1275 3
8 -
=-200.
28.
675 13
6|-r500.
29.
1134 15
10 -
hiooa
30.
4332 13
7f-^3000
44 VULGAB FRACTIONS.
42. To reduce a compound fraction to a simple one.
KuLE. Multiply together all the num" for a new num',
and all the den'^ for a new den'.
Ex.1. fofJ=^.
For one-third of J is ^ (37) ; therefore two-thirds, which must he twice
as great, is ^ (36).
By similar reasoning, f of § « ^ = § of }.
Ex.2. |of5=foff=^.
Mixed numbers must be reduced to improper fractions,
before the rule can be applied.
Ex.3. 2f of5of3j=^off of J«2|il=48|.
Compound fractions may often be reduced hj striking out
factors common to one of the num** and one of the den".
Ex.4. ? of S of ^=? = lJ(35,OB8.).
5
5
-5 \-"-» "•"•"y.
Ex. a7. Express as sim
pie fractions
1.
i off of 4.
2.
§ off of 6.
a. 1 of for 3.
4.
|of|of3i.
6.
1 off off.
6. 1 of 3ft of 9|.
7.
foffof^.
8.
f off of 31.
9. 4jof3|ofl0.
10.
21 off of 71.
11.
foffofrj.
12. 3jofl|of3f.
13.
fofiJofOofei.
14.
^of2|ofl^ofl|l.
15.
A of A off of 7.
16.
1 of 61 of if of ft.
17.
ft of 1| of 51 of 1
18.
llof2|of3|of 4f.
19.
3f of2jof|of^.
20.
ftof2jof?ofl0|.
43. To reduce fractions to a common denominator.
Rule. Find the l. c. m. of all the den", and take this for
the common den'^ : for the new num" multiply each num'^ by
the number obtained by dividing the common den'^ by its
own den'.
Ex. Red»ce f, fl, ^, to their least common denominator.
The L. c. M. of 8, 12, 18, being 72, we have
5_ 5x9 _45 11, 11 x6 _66 7 , 7x4 ^28
a 72 72' 12" 72 72' 18 72 '^72'
VTTLGAR FRACTIOKS. 45
V'here the factors 9, 6, 4, ia the new num'* are obtained by dividing the
common den' 72 by the original den'* 8, 12, 18, respectively,
Fbr, in any one of these fractions, it is plain that its num' and den'
have both been multiplied by the same number, viz. that which makes its
den' =72.
Ex. 28. Kedace to their least common den'
1.
2» 5» 6» f •
2.
e» 7» 5» IT'
Q 2 3 5 7
*'• a» 4» 5» S*
4.
8» 9» 16> T8'
5.
3 7 15
4» 8» l6»
31
32*
C 5 5 2 13
o. e» ¥» 9» 25*
7.
7 11 17 19 85
16» l8» 54» Se» 48*
8.
5» 9» 27> 8T» 54S*
9.
4 3 5 17 4 15
7» 10» 12» 53> 6i» 51*
10.
11 17 5 7 3 86
2T» 54» 6> TS* S» l6*
1.
3 7 6 11 13 33
5* 10* l5* SO' 45' 60*
12.
T» l2» l6 » 2T» SJ» 5o*
44. Addition of Fractions.
Rule. Reduce them (if necessary) to their least common
dcn^ ; and take the sum of the num", retaining the common
den*".
Ex.1. 1 + 1 = 1.
For the whole being divided into 5 equal parts, 3 of those parts, to>
gcther with 1 of those parts, must make 4 such parts.
P.V Q g+a . 4_Jl2±l*±i5_-_133._.o43
XiX. ^. S^4^6~~ 60 60 ^60*
If any of the given quantities are whole or mixed numbers,
it is best to take separately the sum of the integral and
fractional parts, and then add the two results together.
Ex.3. 2| + 3^+5^ + 4.
Here 3 + JL + -5-=2£±^±2?«jti5 ^-iM-iU.
ntrc 5 -f iQ -r 12 60 eo ■*6o~'i2»
.-. 2 + 3 + 5 + 4+lll=15l|.
Improper, fractions should be reduced to mixed numbers,
and compound fractions to simple ones, before the application
of this rule.
Ex.4. ip + |of^ + 2|0f 2^of |+5 = I4i + |-f-3i+fi,
•"cre g-j g-t 5= 2* — in — *24»
.-. 14 + 8 + 5 + lj|.a23^»
46 VULGAB FRACTIONS.
Ex. 29. Find the value of
1.
$ + ? + ? + ? + ?.
2.
i + i<-s+tV
4.
ii+^+u+je.
5.
l+l + S + H-
7.
ft+}|+i^i.
8.
^ + ifi+J- + H.
10.
32 + 2§ + A + 3|.
11.
2i + ? + 4 + 5e.
13.
^+11^ + 2/5 + ^.
14.
T5 + if + 5T + 40-
3. j+i+|+{.
e"^-!. "^ J. ^ -I. Jt
9. 2j + 3j + 4i + 5,
12. l| + i + & + 2A-
15. 3A + ft + li+iH.
16. 17^ + ? + A+l^. 17. f of 18 + f of 1^.
18. H+lft + A + 2U+A. 19. lii + 2|3 + 3U + 4|§.
20. 5j + ?of7j + 8^. 21. | + 7A + tof?of lOj.
22. 22of3§ + ^+2|of4jofli + 4|offtof2|ofl?.
£ 8,
A
£
s.
d.
;£
s.
d.
;£
«.
d.
23.
3 5
7|
24.
7
5
n
25.
3
15
n
26. 7
11
8|
4 10
8i
•
2
13
H
5
14
n
2
9
7A
5 6
5|
5
11
4|
7
6
lOf
6
5
4*
6 12
9|
2
8
5|
8
1
iiii
3
18
7&
7 5
2^
7
17
3A
2
4
6|
4
5
61
2 3
8 5
Of
28.
7
10
13
4^
29.
1
4
5&
3
30.23
19
2
2i
27.
17
13
5i
6|
6 1
2|
2
17
4f
32
6
nj
14
1
5J
5 17
8|
5
2
8|
12
10
9|
7
"8
tl|
6 4
2^
6
11
2M
7
8|
4
9
5ii
5 1
n
4
5
0?
11
5
4i
16
4
2J
7 12
m
6
3
4J
6
16
5^
5
4
3J
45. Subtraction of Fractions.
EuLE. Heduce them (if necessary) to their least common
den'y and take the difference of the num", retaining the
common den^
Ex.1. f-J=f.
For the whole being divided into 5 equal parts, and 1 of those part»
being taken from 4 of those parts, there will remain 3 such part&
VULGAR FRACTIONS. 47
If the given quantities are both mixed numbers, or con-
sist of a whole and a mixed number, it is best to take sepa-
rately the difference of the integral and the fractional parts^
and then add the two results together.
Ex.3. 5f-2|.
Here f-|=-5^*«|; .•.5-2 + |=3|.
Ex.4. 5|-2i.
Here f-l-^s^^ J-; .•.5-2-|=3-l-2|.
Ex. 5. 6-4f=2-?^l$,
Improper fractions should be reduced to mixed numbers
and compound fractions to simple ones, before the application
of this rule.
Ex. 6. Jof2lof 16-lf of5|=8-7f«7J-7f«f.
Ex. 30. Find the value of
1 11—8. 13 T, 8_9. 1_1
^* Is T5» 20~5U» T5 20» 5 5*
2. S|-ll; 3|^2f} 5-2f; 10|-iJ.
8- 1A-|J 9-3Ai 97|-48|; 5^-21?.
4. 13^-3^; 4^,-3i; 3|-^; 24^-21^.
K l.g 4.171, •4.J.8_lnf2. 9—1 nf«
0» *25 7» *^*3S 2T» *6 4 "* 3» 10 5 "* TI*
6. l|of2j-3}|5 5jof4l-3|of3j.
7. 3i + 4f-5i+16|-7|i+10-14f.
8. 51-21-3^ + ^-161 + 3^+81.
£ 8, d, £ s, d, £ 8, d,
9. 13 5J 10. 4 17 11| 11. 9 01
4 17
6J
12.
15
9 19
3f
9^
3 19
4J
la
7 17
6 19
7A
9^
8 17
n
14.
8 13
4 19
6f
91
46. Muliiplication of Fractions,
Rule. Multiply the num" together fw the new num^ and
the den" for the new den'.
Ex.1' |xt=4*
■ 2
48 VULGAR FRACTIONS.
This method is the same as that we should have used to
find the value of the compound fraction f of ^, or ^ of J,
(42) ; and we must here observe that the same word * Multi-
plication' is used to signify, not merely, in its original sense,
and as we have hitherto employed it (when the multiplier
was a whole number), the taking a multiple of a quantity,
i. e. repeating it some number of times, but also (when the
multiplier, as here, is fractional) the taking any pa?t or
parts of it ; so that ' to multiply f ^7 7 * is only another way
of saying *to take J off; and hence the Rule for the oper-
ation is the same in the two cases.
It will be seen, however, that this Rule includes the case of
Mult" by whole numbers ; thus if we had to find the value of
f X 5, we might say, ^ x 5= J x |^ = J*^j =-^-y obviously the
same result as we should have obtained by the common rule
of Mulf* by whole numbers (36) : and it is on this account,
viz. that the general method of taking any part or parts of
a quantity includes the particular case of taking any multiple
of it, that mathematicians have adopted the name, properly
belonging to the latter case only, and applied it also to the
former, calling the operation in both cases multiplication.
The method, therefore, of Mult" of Fractions is the same
as that for reducing a compound fraction to a simple one ;
and (as in that case) mixed numbers must be reduced to
improper fractions before applying the rule, and the result
may be simplified by striking out factors common to num'
and den*^.
Ex. 2. 2| X 3| X If of § of 10=^ X I X I X I X io =308 ^ 102|.
Ex. 31. Find the value of
2. IJx 21x100; 13|x3f xlg^g; 63x2f of2l.
3. 2i of 3| X 4f of IJ; 2} x 1| of 1^^^ x 3l of 1^.
4. lof ^of |xAof3l; Ifoffx Aof2Aof8.
5. fxlfof 12lx2iof^; |of l}x2|of 4| of 2|.
VULGAR FRACTIONS. 49
47. Division of Fractions.
Rule. Invert the divisor, and multiply,
T?-v 1 a_.5_av7 21 ii
Here also the word ' Division ' is used in a more general
sense than heretofore, to denote the finding that quantity,
which, multiplied by the divisor, will produce the dividend
— the word multiplied, being here used in the enlarged sense
explained in (46). Hence, in the above Example, where the
div' is ^ and the div^ f , we must have quotient x f = J :
multiply each of these equals by the same quantity |^, and the
products must be equal; .*. quotient x ^x J=J x ^i but
f X 1^= I ; hence the quotients I x |^= l^^^' ^ above.
The quotient thus obtained will have its usual meaning,
when the div' is an integer, i. e. will express how many times
the div* contains the div*^, or what multiple the div* is of the
div' ; thus f -5-5 = J-i-^= J X i=^i and hence J contains ^
five times or =5 x -^^x but when the div^ is a fraction the
quotient will express what part or parts the div^ is of the
div^ ; thus ^-5-^^= (as above) I^, and hence %^\-^-q of f.
Mixed numbers are reduced to improper fractions, and
compound fractions to simple ones, before applying this rule.
-UJW. AT, ^la -S- «4 — 3 . -4- = 3 * 16 — 45'
Ex.3. (2|of 33)^(41 off of j*5)=10-ff= 10x^ = 35.
Ex.4. i£i==± = ?ix21.126^5^.
Fractions in which the num'^' or den", or both, have a
fractional form are called Complex Fractions. Methods of
simplifying them are in general easily devised.
Ex.5. Simplify ^1 + ^-^^
4A-iof(l-i)
Here iof(l-5'L) = 3of|i = |i; hence
a + ?r-5f - (175 + 80-236)x96 _ 19x96 ^ *I\
♦» _3i /Qoo__9i ^xy^A^ oci^iim\"ic\v.
^\
li-U (3P2-3I)xI00 361x100 \^x1b \lb
50 VULGAR FRACTIONS.
Ex.6. (7i.JofS)x||of_|_,^
^/38 5x26x2 \ 5x24 ^, ^6
\5 9x25x3/ 8(260-3) * 13'
Ex. 32- Simplify
1. 2-J-l; l-i-f ; 2|4.1§; 2^+3|; 16|^121; ff^-^.
2. ll/j+f J 1-5-H; (|of|)+(?off); (4jof^)4-(6|of 1?).
3. 209-i-Jof20; (f of |)^(f of | of 5); (41 of 31)^(21 of 6^).
4 52.3f.Ji.iJ ^ 23 . S^ofll , 3?of2H . 21+1|
6. (3i^|off.|f).||±||.
We shall here give examples of the application of the pre-
ceding rules to the Mult" and Div" of concrete quantities.
Ex. 1. Knd the value of f of £4.
Since (32) | of £4 is the same as | of 3^4 x 8, we first multiply £4 by
3. and then divide the result by 8.
£4
8
8) 12
£i 10 Ans,
This is the same (46) as to multiply £4 by f.
Ex. 2. Divide 1 ton 13 cwt 15 lbs. by l|.
Since I5 =5, wc have here (47) to multiply by |. We may do this as
in Ex. 1, or (which is often more convenient) by first dividing by 2,
Jirhich gives \ of the quantity, and then dividing this half by 2, which
jtres J of it} and adding the two results together, we shall have f of it.
ton cwt. qrs. lbs. ton cwt qrs. lbs.
1 13 15 1 13 15
? for I
4) 4 19 1 17 for I
14 3 l\\Ans,
16 2 1\
8 1 3|
1 4 3 \\\Ans.
\ ,
Compound fractions must be reduced to simple ones before
the application ofthh rule ; but, m the case of mixed numbers^
VULGAR FRACTIONS. 51
it is best to multiply separately for the integral part, and add
the result to that obtained by the Rule for the fractional part
Ex. 3. Multiply £2 10*. 4d. by 3^.
£2 10 4
^
12 )12 11 8
1 11|
7 11
£d 11 Hi
Sometimes it is convenient to reduce the given quantity
to one denomination, before applying the Rule.
Ex. 4. Divide 7*. IJdL by ^.
Here 7«. IJA =342 farthings, which we have to multiply by iJ*-= 12 J.
342/
9 )2394
2G6
4104
4 )4370
l 2)l092l cL
9U 0|</.ȣ4 lU 0|d
Ex. 33. Find the value of
1. f of£l; J§of£5;6*. Srf. x f ; 3f of 2«. 6</,; 2f of 21#.
2. £3 6 8x^; £3 7 5-5- Ij; £5 4 6i-5-lJ.
3. £7 6 8jxl|; £S 7| x 2f ; £l0 112|x3f.
4. £l3 15 4x4f; £l8 17 0x4|; £2 10 6|x3j.
5. £30 14 6i-^^; £7 13 4-5-}f; £4 7 3f+^.
6. f of a ton ; } of a lb. Troy ; 3 cwt. 1 qr.-rl^ ; 11| of 6«. llJA
7. 2 wk. 3 d.-r^ ; 3a. 3b. 3p. x 10^ ; 2s, 9|rf. x Jof 5j.
8. I of ISjj.; 1 cwt. 2 qrs. 13 lbs. x 3^ ; 13|| of £7 5a. lOdl
9. £llU.U + % ^of£8 8«.5ld:;^of^of^of27#.
10. 1 m. 5 fur. 91 yds. 2 ft.-r2| of 1^ ; £3f + 9yV. + 5|(/.
11. ^ + ^. + 1 of 2U ; I cwt + 8| lbs. + 3^ oz. ; 4 d. 5 h. x 1^.
12. l|of 10*. 6<;.-f of 2«. 6rf. + £^-^of2U
la f of21«. + |of 5&-t-|of£312s. 6(1.
52 VULGAR FRACTIONS.
14. f of3I*.-i-}of 5«. + |of 7*. 6c/.-|of2dL
15. 2|of Ifof 82d + 3| of l}5of^of4l<f.
16. f of£l5 + 3fof£l +Joff of|of£l+|of?«.
48. To reduce a given quantity to the fraction of another
given quantity.
Rule. Reduce both to the same denomination ; and take
the result of the former for the num*^, and of the latter for
the den', of the fraction required.
Ex. 1. Reduce 7*. 7J. to the fraction of £l.
Since Is, 7(/. = 91(/., and £\ =240(/., the fraction required is ^.
For 1</. is 5^5 of £i ; and therefore 7*. 7rf., which = 91(/., is ^ of £\.
Any common denomination, to which the two quantities
may be reduced, would answer the purpose of expressing one
of them as the fraction of the other ; but if the highest of
which they both admit be taken, the fraction will be ex-
pressed in lower terms.
Ex. 2. Reduce half-a-crown to the fraction of half-a-guinca.
Reducing them to pence, we have the required fraction =^"g; but
reducing to sixpences we have the same fraction in lower terms, = ^j.
Note, 5Y expresses what is called the Ratio of 2s. 6rf. to 10*. 6(/. ( 79 ).
Ex. 34. Reduce
1. 3*. 4d, to the fr. of £l ; 2«. 6§d to the fr. of 6d,
2. £7 9s, 6d. to the fr. of :£13 4s. 6d. ; 6^. 8|<f. to the fr. of l^.
3. 3 qrs. 14 lbs. to the fr. of 3 cwt. 1 qr. ; 1 ton 4 cwt. to the fr. of
15 cwt. 1 qr. 20 lbs.
4. 3*. r^d, to the fr. of £\ 3.9. 4i</. ; £4 7s. 6|c?. to the fr. of 27*.
5. 3 cwt. 2 qrs. 3 lbs. to the fr. of a ton ; 14 h. 15 m. to the fr. of
3| days.
6. 2r. 13p. to the fr. of 3 acres ; 14 half-crowns to the fr. of 6«. 8c/.
7. A ton to the fr. of 3 cwt. 3 qrs. 21 lbs. ; 30p. 5 yds. to the fr. of
1 fur. 28P.
8. 3 w. H m. to the fr. of half-an-hour ; 3 qrs. 2 qts. to the fr. of
4 qrS' 3 bus.
9. 8 A. 3n. to the fr. of 2a. 32p. ; 1 ft. 2f in. to the fr. of a yard.
)p, 7h' 12 m, to the fr. of a day ; £4 1?^. l\d, to the fr. of £l 9s, 3|d.
VULGAR FRACTIONS. 53
11. 17 lbs. to the fr. of 1 qr. 14^ lbs. ; 1 m. 4 fur. to the fr. of 3 yds.
1ft.
12. 2 sq. yds. 2 ft. 120 in. to the fr. of 3p. ISjyds. 1 ft. 72 in. ; 3 cwt
14 lbs. to the fr. of 2 ton 2 cwt 2 qrs.
13. £22 \3s. S^d. to the fr. of 3l gs. ; £3 168. 6^ to the fr. of
£1 3s, 5i</.
14. 3000 in. to the fr. of 1 fur. 5p.; £2 Os, 3|</. to the fr. of £l 4«, 2}d,
15. 1| guineas to the fr. of £l|; £ll 6«. 5(L to the fr. of £10 58. id.
16. 3| crowns to the fr. of £l 12«. 9|</. ; 2| half-guineas to the fr. of
10<. 11^
49. To reduce a fraction of one given quantity to a frac"
tion of another.
EuLE. Express by (48) the first quantity as a fraction of
the second ; and the fraction required will then be found by
reducing the resulting compound fraction to a simple one.
Ex. 1. Eeduce |«. to the fraction of £1.
Ex. 2. Beduce l^ h. to the fraction of 10 min.
1 h.=fgof 10m.-foflOni.; .M^h. = l^of f of 10m. = 6| of 10 na.
Ex. 3. Beduce 3f of £l 0«. 9|</. to the fraction of £l 10«. lOd
£1 0«. 9|rf.=999/:, and £l 10*. 10rf.= 1480/.;
hence the required fraction s=3| of ^^«2|.
Ex. 35. Beduce
1. £| to the fr. of a guinea ; 1|«. to the fr. of £l.
2. |rf. to the fr. of 15«. ; 12| of 3*. Gd to the fr. of £1.
3. f of 1*. 6rf. to the fr. of 1«. ; f of a sixpence to the fr. of £l.
4. 31 of £1 3s. id. to the fr. of £5 ; 2| of 17«. 6|rf. to the fr. of 10s.
a. 3^ of 1 cwt. 3 qrs. to the fr. of a ton ; 3f d. to the fr. of 3 wks.
6. 1 J of £3 I3s. ed. to the fr. of 10«. 6d. ; 2f of £6 to the fr. of £l 13*.
7. 2| of 4 cwt. to the fr. of 3 qrs. 4 lbs. ; 4| crowns to the fr. of 5 gs.
8. I lb. Tr. to the fr. of a lb. Av. ; | po. to the fr. of a fathom.
9. I sq. ft. to the fr. of a pole ; 12| of 1 qr. 3^ lbs. to the fr. of
1 ton 2 cwt
10. Sjof 2a. 3ii. to the fr. of 2r. 2|p. ; 1^ of £2 48. 1\ii. l^VSwi ^t. o1^»*
64 VULGAR FRACTIONS.
11. 3f wks. to the fr. of Id. Sfhrs. ; 2| of 45 yds. to the fr. of 10 mfles.
12. 2| of 3R. 6p. to the fr. of lA. 2r. 3p. ; f of 1| of 10*. 7ji. to the fr.
13. 33j of 3 qrs. to the fr. of 3| tons ; 3f of 1|a. to the fr. of 2a. 2lp.
14. 7J of £2 Ss, 6i</. to the fr. of 7«. 6<;. ; | of 58. + J«. to the fr. of 21*.
15. 4| of £2 \3s. 7|df. to the fr. of £2 Us, S^d ; If of £2 0«. l^, to
the fr. of £2 2*. 2 J</.
16. 6|f of £\ 10*. 53df. to the fr. of £S S*. Oj^i ; | of £l -| of 21«. to
the fr. of 108. 6d.
11
I6
Miscellaneous Examples. 36.
1. Which is the greatest and which the least of A» &> 55 ^
2. Divide the sum of J, J, and ^ hy the difference between J and J.
3. What n** added to |f makes If ? and what taken from Iff leaves
?
4. Which is the greater, f of 2f, or J of 1 J, and by how much ?
5. Divide the sum of 10 and ^ by the difference, and also the
difference by the sum ; and find the sum and difference of the two
quotients.
6. Divide the sum of f of £3 la. 6d., and | of 4| guineas, by lOf .
7. If I pay away J of my money, then | of what remains, and then
\ of what still remains, what fraction of the whole will be left ?
8. What n« added to ^, ^, ^, g, will make the sum total 3 ?
9. What must be the length of a plot of ground, if the breadth be
15| feet, that its area may contain 46 square yards ?
10. Add together the sum, difference, product, and quotient (the
greater being divided by the less) of £ and ^.
1 1. Find the value of | lb. Troy + 1 oz. Troy ; and of £i~f«.
12. Express 2| ells as a fr. of a yard ; and mult. 3 ft. 7| in. by 2^ in.
13. Add the sum and difference of ^ of 3 guineas and § of £4.
14. Divide '^ (H^^'H^ by iL and find the value of J"''g'^4 ,
V^ fA ^ ill
15. To ^ of a dozen add || of three hundred, and divide this sum by
tlie difference of 3^ of a hundred and 43|.
VULGAR FRACTIONS. 55
16. Multiply the sum of 1, |, |, and |, by the difference of ^ and ^;
and divide that product by the double of 21|.
17. Take from 1 its half, third, and twenty-fourth parts; add the
product of those parts to the rem'; and multiply this sum by 7j|.
18. Multiply the sum of 3|. 4|, and 4|, by the difference of 7f and
5|; and divide the product by the sum of 94| and 93j.
19. Divide 2 by the sum of 2|, f, and 4 ; add If— | to the quotient;
and multiply the result by the difference of 5| and 4|.
20. Find the value of (J + 1) x (1| + 2|) x (2^- li) x (3^-f ); and
of l|-5-2|+5i-i-3i.
21. A person had ^ of a lottery ticket, which was drawn a prize of
of £518 105. • what was the value of his share ?
22. Express the sum and difference of £^ and f of a crown as
fractions of half-a-sovereign ; and find how many times the first contains
the second.
23. Multiply 15f*. by 109f, and divide £61 4s. 7^. by 267^.
24. How often is fs. contained in half-a-crown ? and how often is
£\ contained in 24 guineas?
25. If a yard of lace cost £l||, what will 16|J yards cost ?
26. If f of a ship be worth £3740, what is the value of the whole?
27. Compare, as fractions of their highest common denomination, the
values of ^ of £l, ^ of a guinea, and ^ of a crown.
28. Find the value of ,,, ^/,t^ , , x ? of ii^lii.
(Ifoff)-lOi 6 13iof6l
29. If I of an estate be worth £220, find the value of ^ of it.
30. Express in Tr. weight the difference between | lb. Tr. and f lb. Av.
31. Find the value of (12|-8|-l^ + ^)x4|x (7^-6j), and of
32. Compare, as fractions of their highest common denomination, the
values of ^ of half-a-crown, ^ of 3s. 4</., and ^ of 4s. 2^
33. Express, as a fraction of £5, the difference between £7f and
£7 x I ; and find the value of £l4lf -^1^?.
34. A person owes a guinea to each of 4 creditors : to one he pays
\ of bis debt, to another f, to another |, and to another |§ ; what will he
still be owing altogether?
35. Expresfir ifl Troy weight the sum of 3|\V)8, Tt., aa^ \^\\\>^. K:'*
66 VULGAR FRACTIONS.
36. Find the value of Hll^of iLt^^ of ||lii.
87. If ^ of a ton k worth £4 lOs,, what is the value of J of it ?
38. Ailcr taking out of a purse f of its contents, | of the remainder
was found to be ISs, 5^. ; what sum did it contain at first ?
39. The dimensions of a room are 29| ft. bj Uj ft. ; what length of
carpet, {yd. wide, will cover it? and what will be the expense of it, at
d^s, per yard ?
40. A ship is worth £16000, and a person, possessed of ^ of it, sells
f of his share ; what share has he remaining, and what is it worth ?
41. Express 4 bus. 1 pk. 1 gaL 2 qts. as a fr. of a qr. ; and reduce
5 cwt. to lbs. Troy.
42. If i of a ship be worth £36 10«. 7|i., what share will cost
£125 58, ?
2 2^
43. Multiply 3^ by 15f, and divide _ by -^; and add together the
3j 3
sum and difference of these results.
44. A party having a bill to pay of £12 7«. Ijcf., one of them pays
for himself and three friends the sum of £5 9«. lOJ. ; how many were
they?
45. Simplify, and then add,^ll^^i-^ end ^ ^''^- ^ ^^
46. A pint contains 34| cubic inches ; how many gallons of water
will fill a cistern 4 ft. 4 in. long, 2 ft. 8 in. broad, and 1 ft. 1| in. deep ?
47. Add together If, 2|, and 3|; multiply this sum by the product
of these fractious ; subtract from the result the difference of 2| and 1| ;
and divide the remainder by the snm of 5^ and 1| of 3J.
48. How many yards of paper, f yd. wide, will be required for the
walls of a room that is 20f ft. long by lljft. wide, and 12|ft. high?
and what will be the cost of it at 2^, a yard ?
49. A cubic foot of wood weighing 1 1 j^lbs., what is the weight of a
beam 24 ft. long, 22 a wide, and 2ift. thick ? and what is its value at
3jf5. per cubic foot ?
50. A person dies worth £10000, and leaves J of his property to his
wife, i to his son, and the rest to his daughter. The wife at her death
leaves f of her legacy to the son, and the rest to the daughter ; but the
son adds his fortune to his sister's, and gives her J of the whole. How
much will the sister gain by this ? and what fraction will her gain be of
tAc whole?
57
CHAPTER IV.
DECIMAL FRACTIONS.
50. In common numbers, or decimal integers, the actual
value of each figure depends upon its position with respect to
the place of units, its value in any one position being one-
tenth of what it would be, if it stood one place further to the
left : thus 3045 denotes 3 thotisands, hundreds, 4 tens, and
5 unitSy or 3000 -f + 40 + 5; where we may obtain the
actual value of any figure by multiplying it by 10, 100, 1000,
&c., according as it stands in the 1st, 2nd, 3rd, &c. place to
the left of the place of units.
Now if we continue the same method of notation to the
right of the place of units, still reckoning the value of each
figure to be one-tenth of what it would be, if it stood one
place further to the left, we obtain what are called decimal
fractions, or briefly decimals ; thus setting, as is usual, a dot,
called the decimal point, after the unit's place, the number
3.045, &c. will denote 3 units, tenths, 4 hundredths, 5 thou^
sandths, &c., or 3 + -^^ + ^^ + y^*^^^ + &c. ; where we may
obtain the actual value of any figure by dividing it by 10,
100, 1000, &c., according as it stands in the 1st, 2nd, 3rd,
&c. place to the right of the place of units.
51. Hence it follows Ihat a decimal may also be defined to
be a fraction, whose den^ is 10, or some power* of 10, as 100,
1000, &c., which den^, however, is not set down, as in vulgar
fractions, under the num', but expressed by marking off by a
point, from the right of the num', as many figures as there
are cyphers in the den', prefixing cyphers to the former, if
necessary, to make up the requisite number of figures after
the point.
* A power of a number is the product of a number multiplied hy
it:- elf once or successively. When the number is used qa at IwiVat VwSRfc,
thrice, &c, the product is called the second poTret, XYivt^ ^qti^x^^^-^^'^
the n*.
58 DECIMAL FRACTIONS.
1000 1000 l*'0 1000 *
2125_O.10/> ,119_— .0110 _-2I__— '00027 &■(*
iobo""^^^''' loooo" ^iiVf 100000 uwu^/, o^c.
) 52. Conversely, any decimal may be expressed as a vulgar
\ fraction by setting down the figures which compose it as the
^ num*", and for the den', 10, 100, 1000, &c. according as there
are one, two, three, &c. figures after the point. This, in
fact, amounts to expressing each figure separately as a vulgar
fraction with its own den'', and then bringing all these frac*
tions to one common den^
Thus 2-03 = 2^ or fga ; 'Z79^{^ + ^ + rh>'-^^^-mi
42037-42j§j5or^=^^; .0029 = ^o'oo; 16001 rrlS^^ or .^^
Sometimes the resulting fractions admit of reduction to.
lower terms.
Thus 13-75 = 13^^ = 132; 23-0625 =:235§|-J5=23-,V
53, Any decimal is multiplied by 10, 100, 1000, &c. by
moving the point one, two, three, &c. places to the right, and
divided by moving it similarly to the left.
Thus
3 247 = fia? ; hence 3-247 X 10 = -Y^= 32-47; 3-247 4-10=^0^0 = -3247.
3-247 xl00«2|42«324-7; 3247 + 100 =j22it_ ^.03247,
So '0023 X 100= -23, 2-3-!- 100= '023,
2-3 X 1000 = 2300, 23 ^1000 = -0023, &c.
54. It should be carefully noticed, that adding cyphers to
the right of a decimal does not alter its value ; thus *d, *d0,
•300, are all equal, representing each of them ^y or as in (52)
fV TTHTJ TTns% respectively; but prefixing cyphers to the left
of a decimal after the point is equivalent (53) to dividing it by
10, 100, &c. ; thus -3, -03, -003, are respectively y^, y^, ^\j^.
Ex. 37. Express as decimals
1 JL .117 .S3_ 1015 Q ^1 21_' 117 . 3
*• io» 10 » 100' iTioo* *• ioo» 1000?! lo^co' looooood*
3. 2 tenths + 3 hundredths + 37 millionths.
4. 11 tenths + 11 thousandths + 11 hundred-thousandths.
5. 13 + 3 thousandths + 5 millionths.
DECIMAL FRiCTIONS. 59
6. 101 tenths + 10 thousandths + 101 millionths.
Express as vulgar fractions
7. -037, -0002, -25, -375. 8. '0075, 1-226, -1875, 3*225.
9. -0006875, -0009375, 23038125.
10. 15-203125, -00234375, 40078125
Multiply and divido
11. -3 by 10 and 1000, -00125 by 100 and 10000, 538*734 by ten
thousand.
12. M by 1000 and 1000000,11-025 by 1000 and 100000, and
213-012 by a million.
55. Addition and Subtraction,
Rule. Set down the decimals with their points in the
same vertical line, so that units of the same kind may ho
under one another, filling up the blank places with cyphers;
then add or subtract as with common integers, setting tlio
point in the result in the same line with the other points^
Ex. 1. Add together 28146, -0938, 8,' -875, 31,2788, 4-0087.
28 146 Here the figures in the right-hand column represent so
-0938 many ten- thousandths ; so "we have to add together
8*0000 6+ft+0+o+S+7 29___a_.__2 .
*8750 10000 ~~ 10000 ~ 1000 "*" loodo »
31 '2788 yfQ get down therefore the 9 under the column of ten-
4-0087 thousandths, and carry the 2 thousandths to the next
' * column ; and so on.
Ex. 2. Find the difference of 2-418 and 1-2234.
Here we have 4 ten-thousandths in the lower line, but
1-22^4 none in the upper; we therefore have to borrow one from
the 8 in the next column, i.e. we borrow 1 thousandth
s= 10 te7i'thousandthSt from which we take the four ten-
thousandths, and have 6 remaining ; we have now only 7 thousandths
in the upper line, from which we are to take 3 thousandths, or, instead
of this (as in former cases of borrowing in Subtraction), we may take
4 thousandths from 8 thousandths ; and so on.
Ex. 38. Find the value of
1 . 1 1-275 + -34132 + -00414 + -0001 + 23001.
2. 321-4 + 12 + 31-6154 + -01 + 2214 + 415*62.
3. -001 213 + 45-613 + 234 + -0012 + 141-00056.
4. 1-0000123 + 31-1 + 117-154 + 2343008 + «0002.
6. 32001-12 999^; and 3-45-*00098.
6. 231415-2-008; and 8-412-2-99987.
60 D£C1MAI< I^BACTtOKS.
7. 220001~2'9999 ; and 24156- 2414-5987.
8. -001 --0009987 ; and 24004--987616.
9. 1-3742 --03742 ; and 3054- -3054.
10. -0123- -009087 ; and 333 -2-98765.
66. Multiplication.
EuLE. Multiply the given decimals as if they were com-
mon integers, and mark off in the product as many decimal
places as there are in the multiplier and multiplicand
together,
Ex. 1. Multiply 1-0026 by 2-6.
1-002 6
2-5
5012 5 For 1-0025 x 2-5 = igg§§ xf|-|g§§-2§ = 2-60625.
2 0050
2-5062 5 Ans,
Ex. 2. Multiply -0048 by -000012 ; and 1-006 by -006 x -0064.
•0048 1-005
•000012 -005
•0000000576 •005026
•0064
20100
30160
•0000321600 :=r 00003216 Jns.
Ex. 39. Find the value of
1. 22-5 X 32-16 ; and 441 x 3321.
2. -0001 X -001 ; and 32 1 x 2-31.
3. -0032 X 23*45 ; and -0002 x 301.
4. 22-5 X -0241 x -0024 ; and -0003 x -01 x 500000.
6. 2-7 x -27 X -027 x 270 ; and -2 x -04 x -008 x 64000.
6. 11 X Oil X 1^01 X -0101 ; and -013 x 16 x -007 x 305.
57. Division.
Rule. If the given divisor is not a whole number, make
it so by removing its decimal point altogether, and shift the
decimal point of the dividend as many places to the right
as there were decimal figures in the divisor ; annexing for
this purpose decimal cyphers, if necessary, to the dividend.
DECIMAL FRACTIONS. 61
Then divide as if the given decimals were common
integers ; and when, in the process of division^ the decimal
point of the dividend is arrived at, place a decimal point in
the quotient.
Decimal cyphers may be annexed to the dividend, to any
extent that may be wanted for carrying on the division (54).
Ex. 1. Divide 27753 by 12 ; also -27753 by 12; and 1037 by 305.
1 2'i277-«;9ftn Here tlie divisors are all integral, and the position
— of the point in the quotient is very simply deter-
i9\.n77coArt mined. In the first sum, we take the 12th part of
^,^ ^ , 27 tens, "which is 2 tens and 3 over ; then the 12th of
'0231275
^n';MnQ'r.A/">. ^7 units is 3 units and 1 over; then the 12th of
•^ g, c ^"^ 15 tenths is 1 tenths &c. ; so that the point in the
" ,22q ' quotient comes exactly under that of the dividend.
1220 1-° ^^^ second sum the 12th of 2 tenths is tenths;
' the 12th of 27 hundredths ia 2 hundredths, and 3 over,
&c. ; and here the student should particularly observe, that when the
divisor is a whole number, there will always be a quotient figure, though
sometimes, as here, a cypher, for every decimal figure of the dividend.
Ex. 2. Divide -805 by 2-3, -001029 hy 1-08, and 1 by -007.
2-32805 1-68 )001029
23)8-05(-35 168) -1029(0006125
69 1008
115 210
115 168
420
-00 7)1-0000 336
7 )1000-000 840
142-857 &c. 8^2
In the first of these sums the divisor, 2-3, is mult^ by 10,
which removes the point, and the dividend is also mult^ by
10, by having the point shifted one place to the right. In
the 2nd sum the divisor and dividend are mult^ by 100,
and in the 3rd by 1000, to make the divisor integral. In
the 3rd sum the quotient will not terminate, but, by annex-
ing cyphers to the dividend, we may continue the quotient
as far as we please.
Obs. An integral divisor ending with cyphers may be de-
prived of the cyphers, if we shift the point of the dv^\dftsA
one place to the left for every cypher wilYidx^bVctix \\i\xa>
•45 -r 60 « -045-^6.
62 DECIMAL FRACTIONS.
A little consideration will enable us often to avoid the trouble
of counting the decimal places of the dividend and divisor.
Ex. 4. Divide 15-95 by 275.
2-75U5-950<'5-8 Here, without counting, we may set at once the
13 75 point after the 5 in the quotient, because it is plain
2 200 ^^^ ^^® divisor, which is a little greater than 2,
2 200 ^^ S^ about 5 times in the dividend, which is a
little greater than 15.
Ex. ftO. Find the value of
1. 16-625 -r 2-6; and -0166254-25.
2. 1562-5-*- -00025; and 1-5625 + 26000.
3. 181-3+-00037; and 171-99-*- 273.
4. 9-065-^-049; and 03 -r 001.
5. 8^-002; and 37-5 + 7*68.
6. 15-^6-25; and 17*28 -f- -0144.
7. -00128-^8•192 ; and 17084592+ -00024,
8. -0002 + -0163; and 4-r 00265.
9. 11-1+32-76 ; and -0123 + 3-21.
10. 2117+-0073 ; and -032+2-137.
58. To reduce any fraction to a decimal.
Rule. If the den'' be 10, 100, &c. we may at once express
it as a decimal (51): in other cases, if 10, 100 &c. be 2k factor
of the den', divide the numerator by it as in (53), and then
divide the num' as it now stands by the remaining factor as
in (57), and the result will be the decimal required.
Ex. i=. 121 = .0025; |I = ?-? = -4625.
400 4 80 8
59. Sometimes the division will not terminate, but the
same figures will be repeated over again continually.
T7 "D J 95 9-5 3 '03 J 4 ^ J . , {
Ex. Eoduce — - or — -, — — or -— -, and -, to decimals. t
90 9 1100 11 7
9) 9-50000 11) '03000000 7) 4-0000000
1-05556 &c. - 1§. -00272727 &c. = ^. -571 4285 &c. = J.
Decimals of this kind, in which the same figures are con-
tinually repeated without end, are called Circulating ^ Re
peatingy or Recurring^ Decimals ; and the part repeated is
called the Period or Repetend.
DECIMAL FBACTIONS. . 63
It is usual to express any circulator by writing it down to
the end of the first period, and setting dots over the first and
last figures of the period ; which dots will, of course, be on
adjacent figures, when the period consists of only two figures,
and will coalesce into one dot, when the period consists of
only one figure.
Thus the above results would be written 1*05, *0027, *57142d.
A pure circulator is one in which the period begins im-
mediately after the decimal point; all others are called
mixed,
Ex. ftl. Reduce to decimals
I 2 . 13_. 42 . 1000 Q 106, T 1 17 . 4000 . g8
*• S0» 550 » ^^» eiS • ^* l25» ^'l2S0» 256 > *'T6«
<l 715 . JJ^ . 1- . 1 1 53 A 1 . 1025 , 13 . 7
"• '65> t5S» Sioo> **M250' *• ST2» 102? » Ieoo> FISS*
5 — iof — - • 7iof— i5— • l-2-oflJlof5
16 62* * 2 "' 62500 » *19 "' *Tf "* f*
60. Any fraction, to be expressed as a decimal, should
first be reduced to its lowest terms ; and then, if the den^
contain only powers of 2 and 5 as factors, it may be reduced
to ?L finite or terminating decimal.
For, in reducing a fraction to a decimal, we set a point after the num%
and annex cyphers to it, until the den^ will, if possible, exactly divide it.
Or, leaving out of consideration the point, (which, it is plain, does not
affect the division, but only determines the place of the point in the
result), this amounts to mvdtiplying ike num^ b^ such a power of 10, as will
make it contain the den^. But now, since the fraction is supposed to have
been originally in its lowest terms, the den' can have no factor in
common with the original num' ; if, therefore, it be exactly contained in
Che num' as it now stands, that is, with the annexed cyphers, it can only
be by its being contained in that power of 10, by which the original
num' has been multiplied. But, since 10 contains only the factors 2 and
5, any power of 10 will contain only powers of 2 and 5 ; and, therefore,
the den', in order to be contained exactly in some power of 10, must bo
made up only of powers of 2 and 5 as factors. In this case the division
would terminate, and the decimal be finite; but not so, if the den'
contain any other factors, such as 3, 7, 11, &c., since then no power of
10 whatever would contain the den', nor, therefore, would the original
num', whatever be the number of cyphers annexed^ h^eoxci^ ^tak2Cc|
divisible by it
v2
64 DECIMAL FRACTIONS.
61. If the den^ of a fraction, in its lowest terms, contain
any other factor than powers of 2 and 6, the fraction may be
expressed as a Circulating Decimal, where the number of
figures in the period will be less than the den^
For since, in the division, the figures to be taken down are
always the same, viz. cyphers, it follows that^ whenever we
have dkiij former remainder repeated, we shall also have the
same series of figures repeated in the qiwtient: but, if we go
far enough, we cannot help having some former remainder
repeated ; for, all the remainders must, of course, be less
than the divisor (or den^), and so the number of different
remainders must be less than the den^ itself.
Ex. 1. Kedace f to a decimaL
V) 6*0 ('857142 jj^^g ^^ jjj^y^ y^ in order the remainders 6. A,
^ . 5, 1, 3, 2, which are all there are less than the
^^ divisor, 7 ; the next remainder must therefore be
— . one of these again, and accordingly we find it to be
^^ 6 ; now, since the same figure, 0, is taken down to it
49
as before, it is plain that the whole series of figures
^^ in the quotient will be reproduced in exactly the
— -- same order as before.
^" In the above Example, aU the possible remainders
have occurred, and the period, consequently, consists
of as many figures as it possibly could, viz. one less
than, the den"" • this, however, is not usually the case.
20
14
5
Ex. 2. Reduce |f =3j^ to a decimal.
22) 3-0 ('136
22
80
66
140
132
8 Ans. 3'Idd.
Sometimes a decimal of very long period may be carried
out easily to many places, as in the following example :
DECIMAL FRACTIONS. 65
Ex. 3. Keduce ^ to a decimal.
19) 100 (-05263 Hence ^ = -062634, /.^^ -15789^;
95_ and hence ^ = -0526316789/5 ;
60 .-.^^ -473684210111 = -4736842105^;
38 and hence ^ == -06263157894736842105^,
120 and, by continuing this process, we obyiously double at
^^4 every step the number of figures obtained.
60 This decimal, it will be seen, circulates after the
*^7 eighteenth figure ; so that
3 A=- -052631578947368421.
Ex. ft2. Keduce to decimals
•I 13 . lOri . J129_ . 17 O 41 . Ill . 22 . OQ 52
*■• 9 » 180 » 65 » 13Y5- ^' 14 > ~22~ » "1665 » •^"333*
o 89 . _i2JL . 1 7 e*oi . jtLLL 4. 135 . 29i_ . sts . 1139
"• 9999 » 21 » *• 49500' 33300- ^* 3700 » 29G0 » 925 » 65555*
;; 1 • i_. 1 • l_
*'• ly* 23» 59» 5l*
62. 7b reduce a pure circulator to a fraction.
Since |= -111111 &c., it follows that f = -2222 &c., |^-5555 &c. ;
SO that any pure circulator, having one figure in the period,
may be expressed as a fraction with that figure in the num*",
and 9 in the den^
Again,
5^=i-^ll=010101 &c.; hence g^g = -060505 &c. ; || = -232323 &c.;
so that any pure circulator, having two figures in the period,
may be expressed as a fraction with those figures in the
num*", and 99 in the den*".
In like manner, since
5|5=|-f-lll = -001001 &c., ^ = i^llll = -0001 &c.,
and so on, it will follow that ani/ pure circulator may be ex-
pressed as a fraction with the period itself in the num**, and
in the den' as many 9's as there are circulating figures.
Thus *o78 = g99 = 3y, '0378 = 5yg5 = xjj[i, *00037o =» g^aygg = s^gj.
63. To reduce a mixed circulator to a fraction.
If we had a pure circulator with any figures before the
point, we might either keep these to form a mixed number .»
as 3-4=3|, 5*4S=5|f ; or we might W\ii^ \\iei^V^^ ^"^
once to an improper fraction, with tlie ^aiae detf tvs^\>ei^<3rc^'i
66 DECIMAL FBACTIOKS.
by writing for the num'' all the figures to the end of the first
period, subtracting, however, the figures before the point ;
The reason of this method may be thus seen :
4 3x9 + 4 3(10-1)4-4 30 + 4-3 34-3
89
43 5 (100-1) + 43 543-5 ^^
^9" 99 99 '
Now, if the point be not immediately before the period,
as in these examples, but moved towards the left, this is
equivalent to dividing the decimal by 10, 100, &c., and we
must therefore annex to the den*^, as found by the preceding
Rule, as many cyphers as there are figures between the
point and the first period :
, 34-3 31 ...A 543-5 538 269
thus •034=- = ; 543= = — = .
If there should be any figures of a mixed circulator still
left before the point, it will be best to leave these as they
are, to form a mixed number :
thus 2-46=2^=2|§=.2X, the same as ?1|^=^==2J§.
The above results may be thus stated, as a Rule for
reducing ani/ circulator to a fraction :
Consider only the figures (zfter the point ; then
For the num% write the decimal to the end of the first
period^ subtracting from it (if any) the figures which do not
circulate ;
For the den% write as many ffs as there are figures
circulating, followed by as many 0*s as there are figures not
circulating. See Notb VL
Ex. ft3. Beduce to fractions
1. -S ; -05 ; -54 ; •?2d. 2. -024 ; -0432 ; -00675 ; 2-0455.
3. 3-41S; 0445; ll45; -00440. 4. 40631; 7*6531; 2*345; -09318.
6. 2-OdOd; -64950; 1-0428671. 6. 2-6428671; 619316; 11-28^
64. It may be noticed that, according to the above rule^
^Ae circulator 5 = f = 1 . It is true, "we cannot reverse this
DECIMAI. FRACTIONS. 67
operation, and reduce 1 to the decimal '999 &c. ; yet it will
be evident, hj repeating the period, that this decimal reallj
differs from 1 by a quantity so small as to be absolutely
insensible : thus
where we see that, by repeating the 9's, the difference be-
tween 1 and the corresponding decimal becomes less and less,
and thus may be made as minute as we please, and will at
length become absolutely insensible.
It is in this sense that 1 is said to be the value of the
circulator % and, indeed, that awy vulgar fraction is assigned
as the value of an^ circulator ; so that, in fact, the equivalent
vulgar fraction for any circulating decimal is that to which
the value of the decimal will become more and more nearly
equal as we repeat its period, and from which it may, by
such continued repetition, be made to differ by a fraction as
minute as we please, and altogether insensible.
Whenever, therefore, in a decimal we find the figure 9
circulating, we may at once get rid of the period, by adding
1 to the figure preceding it : thus '4999 &c. = *5, the same
result as we should obtain by the Rule, since
90 90 10
65. Arithmetical operations in which circulating decimals
are concerned, may often be performed, with sufficient accu-
racy for all practical purposes, by repeating the period as
often as shall seem upon consideration necessary to ensure the
result being correct to some given number of decimal places.
Ex. 1. Add together 13-5, 2026, 111'0004, 3-14i5D, 2024 correctly
to 6 decimal places.
2-02525252 H©re, by carrying out the decimals to 8 places, we
1 11*00044444 ensure the accuracy of the first 6 places ; for, although
314159159 the last two are incorrect, and would be altered, if we
2*02402402 carried on our periods farther, yet a little consideration
1 3 1 '74686812 will show us that the sixth and all the preceding figures
will not bo altered, however often we may repeat tfcift -^wio^^.
G8 DECIMAL FBACTI0N8.
In such a case it is generally sufficient to cany out the periods to
three decimal places more than the number required to be accurate,
Ex. 2. From l*02S4l take •62§, correctly to 6 decimal places.
1-023413413
•628888888
•394624625 Ans. -394624.
It is sometimes convenient to reduce the circulators to
vulgar fractions, especially for the purpose of multiplying or
dividing one circulator bj another, in which case the fraction,
resulting from the multiplication or division, may be after-
wards reproduced in the decimal form.
Ex.3. -25 X -36 = 11x11=^= -064; .le^-OOSt^Jg+^J^^fiffi-
611.
Ex. ftft. Find the value (correct to 7 places of decimals) of
1. -136 + -14286? + 2-416 + 2*06 + 42-65 + -008497135.
2. 37-23 + -26 + l-n + -29? + 3-975 + 8 + 4-76 + 740367 + 32-41.
3. •5--09;and-04--00?6923S. .4. 7-6142867; and -042- -05$.
6. 37-25 X -26 ; and 7-?2 x -^97.
6. 3-975 X 8 ; and 74*0367 x 476.
7. '5-r09; and -04^-769230. 8. 7-i- '142867; and -042 -^ -036.
66. To find the value of any decimal of a given quantity.
Rule. As in common Reduction, multiply the given
decimal by the number of units of the next lower den° which
make one of the given den^ : the integral part (if any) of
the result will be so many units of that lower den°, and the
fractional part may now be reduced in the same manner to a
lower den" ; and so on.
Ex. 1. Find the value of :e-36875.
•36875 or, omitting useless cyphers, '36876
20 20
7-37500 7-37600
12 12
4-50000 4-600
4 4
:J-00000 2-0 Jns, la. 4J(I.
DECIMAL FRACTIONS. 69
If the given quantity be expressed in more than one den°y
it should be reduced to one, before applying the Bule.
Ex. 2. Find the value of -07 of £2 10s. ; and of 7366 of 6«. Bd.
Here £2 lOs.-dOs,, and 6s, Sd.=>BOd.
•07
50
•7365
80
3^60
12
68-9200
Ans, 4s. I0'92d.
Ans. 3«. 6d. 6*0
Ex. 3. Find the value of -1 77083£.
•17708333
20
Or thus : -17708^
20
3*54166660
12
3-541661
12
6-4999992 =» 6-5
4
as in (64).
6-50000
4
Ans. 3«. eid. 2^0 20
'9
But it is often best to convert a circulator entirely to
a vulgar fraction in such a case, and so find its value.
Ex. 4. Find the value of 3-2? of a ton.
Here 3*2? = 3^ ; and Z{^ of a ton = 3 tons 5 cwb. 2 qrs. 6 lb. 3,^ oz.
Ex. 45. Find the value of
1. -46 of £1 ; ^68125 of £1 ; and 2325 of £l.
2. 32-6 of 55. ; 185 of 3s. id. ; and 2 375 of 13^. id,
3. -13125 of £o , and -001953125 of £iO.
4. 3-45 of 5 guineas ; and '325 of 1 J ton.
5. 23-42 of a day ; and 1-46875 of an acre.
6. 2-74 of 128. ed. ; and 2225 of £2 2s. Qd,
7. 3-225 of 2i guineas ; and 22-75 of £5 10s. Od.
8. 3-03 of 10s, bd. ; and -0474609375 of £10 Us, id.
9. -176 of 1 fur. 36 p. 2 yds. 5 in. ; and -22 of 3 qrs. 15 lbs.
10. -2775 of 1 sq. yd. 3 ft. 72 in. ; and 32 156 of 3 m. 330 yds.
11. 2-441 of £32 Os, i\d. ; and 33-25 of £3 12s. i\d.
12. 44-045 of llj^. ; and -5s. + '7 of a crown + •125£.
13. •634375£ + '025 of 25s. + -325 of 30s.
14. 8-71875 of 8<?. + l- 146875 of 6s. 8<?.--0625 of a guinea.
15. -375 of a guinea + -1875 of a crown + '3 of Is, 6c?.--875 of 2aL
16. 3'8d of 4s. ; and 610 of 2s. 9|(^.
70 DECIMAL FRACTIONS.
17. 23-45 of 3 m. 5 fnr. ; and 13-2?6 of 5a. 2b.
18. 2-20? of £Z 98. Hd, ; and 2-145 of 6s. 8K
19. -397916 of £i ; and •4097S of a guinea.
20. •571426 of a qr. ; and -285714 of a cwt.
67. To redtice a given quantity to the decimal of another
given quantity.
Rule. Begin with the term of lowest den" in the first
given quantity, and reduce it to a decimal of the next higher
den° ; prefix to this decimal the term (if any) of this higher
den", which is found in the first given quantity, and reduce
the result to a decimal of the next higher den" ; and so on,
until we have thus brought it, if possible, to the decimal of
the second given quantity.
Ex. 1. Keduce £3 17s. 6|c?. to the decimal of £5.
4) 3-00 Here we first reduce 3/ to a decimal of a penny, by
12) 6*7500 dividing by 4 ; the result is -75, i.e. 3/. = -76^., and, pre-
20) 17-O62500 fixing the 6^., we have now 6-75c^., which we reduce to
6)~¥^78r2o the decimal of a shilling; and so on.
-775625 Ans,
Sometimes, as in common Eed action, we cannot thus pass
directly, through different successive den°*, from the first to
the second given quantity ; and then it will be necessary to
express the first as a fraction of the second, and then to
reduce this fraction to a decimal.
Ex. 2. Ecduce 2s. 9|<f. to the decimal of 7s. 9|d!.
2s. 9f c?. 135 farthings 9 25) 9*00 (-36 An»
^^^® 7s. 9|rf. - 375 farthings" 25 75
1 50
1 50
Ex. 46. Reduce
1. 9s. 6rf. to the dec. of £l; and 2s. 1\d, to the dec. of £5.
2. 5s. to the dec. of 13s. 4(/.; and 17s. 3(/. to the dec. of 10s.
3. £\ 2s. 6<f. to the dec. of £1; and 2s. 7Jdl to the dec of 10«.
4. 3s, 3|d to the dec of £l 6s. 6d ; and £3 4s. 2(f. to the dec. of 2«. Ad.
DECIMAL FBAOTIOKS. 71
5. 6s. 6|dL to the dec. of a guinea ; and 78. 10|d^ to the dec. of £2.
6. 9 oz. 2 dr. to the dec. of a lb. ; and 3 far. 33 yds. to the dec of
a mile.
7. 2 m. 1100 yds. to the dec. of a league ; and 12 h. 55' 21'' to the
dec. of a day.
8. 3 qrs. 3 lbs. 1 oz. 7 drs. to the dec of a cwt ; and 18| days to
the dec of a year.
9. 158, 6f<f. to the dec of £4 ; and 1 cwt. 3 qrs. 7 lbs. to the dec of
2| tons.
10. 3f gs. to the dec. of £100 ; and 4| lbs. to the dec of 3 qrs. 12 lbs.
11. 138. Ad. to the dec. of a crown, and 2 tons 4| cwt. to the' dec of
1 ton 11 J cwt.
12. 3| in. to the dec. of \ mile ; and 22 guineas to the dec. of £25.
la 2b. 4p. to the dec of iB. 5p. ; and £2 \\s. 6fd to the dec. of £a
14. 8 sq. ft. 20 in. to the dec. of 12 sq. in. ; and 7«- 6|e/. to the dec
of£l.
15. 2 w. 6|d. to the dec. of 4 d. 3 hrs. ; and £6 12^. 6|d to the dec.
of \\ guinea.
16. 3 hrs. 3' 2^' to the dec. of a day ; and £24 12«. ^)d. to the dec
of £4.
MISCELLANEOUS EXAMPLES IN DECIMAL FEACTIONS.
1. What Tulgar fraction is equivalent to the sum of 14*4 and 1*44
divided by the difference ?
2. What is the value of '0333 &c of half-a- crown multiplied by -5?
3. The circumference of a circle = 3-1416 times the diameter ; find
the radius of the Earth, whose circumference is 24857 miles.
4. If the length of the year be taken at 365|- days instead of
365 '24§ days, its true value, what will the error amount to in four
centuries ?
5. Keduce ^ and ^se ^ decimals ; 3*76 and 3'76 to vulgar frac-
tions ; and multiply -235 by '0021 and 1*2.
6. Eeduce 7«. Qd. to the decimal of £1 ; find the value of £2-6625 ;
and, if 1 oz. cost -031 25£, what will '0625 lbs. cost?
7. Find the value of -eii + -3 125«. + -2 of a guinea.
8. Eeduce ^ and 4^ to decimals; '01 2d to a vulgar fraction; and
divide 18-073 by -0341 and 5300.
9. Find the value of •453126£ + 1*1484^758, -v -1\%*l^a,
72 DECIMAL FBACTIONS.
10. Bediice S75£ to the decimal of a guinea ; and 125 of 3*675:^
to the decimal of 10*55.
11. Find the value of -3006^4 of a day ; and of -OiySST^S of 2a.
12. Find the value of 3§ + 4i+lJi+3J^ both by vulgar fractiona
and by decimals ; and show that the two results coincide.
13. Find the value of 1*875 guinea + 1*875 crown + 1-876 of 3 625£.
14. Find the difference between 5^ half-guineas and 3*125£; and
roduce the result to the decimal of half-a-crown.
15. Multiply Is. 7H by 5782*5 ; and divide £168 55. 4^. by 1*32.
16. The price of J an oz. of coffee is •4583». ; what is the value of
•0015625 of a ton?
17. Find the difference between 1*6 of 3 4 of 1*125£ and | of 3*6 of
9*1 125;^.
18. Reduce ^^ and ^ to decimals ; "0675 and '0676 to vulgar
fractions ; and find the value of '73125 of £5.
19. If a lb. of sugar cost -0703125 of 85., what is the value of
•0625 cwt. ?
20. Add together |, |, ^ and ^, both as vulgar fractions and as
decimals ; and show that the two results coincide.
21. Find the value of 3-55. + 2*9 of 233755. - ^ of 1 6*65.
22. Find the difference between 17-428571 sq. ft. and 1008 sq. in.;
and between 1*76 cub. yds. and 26*66 cub. ft.
23. Multiply -0235 by 808; divide -0625 by 2*5; and find the
value of -8435416 of £5.
24. Multiply 65. O^d. by 85*3125 ; and divide £10 ll5. Sd. by 29-25.
25. rind the value of 44 of a guinea— 3*75 of half-a-crown + •4l6£
— -3571428 of a guinea.
26. How many yards of matting, 2*4 feet broad, will cover a floor
that is 27*3 feet long and 20*l(j feet broad?
27. Find the value of -375 of 5*375£, and of -06328125 of £100;
and reduce £2 75. 9|i. to the decimal of IO5.
28. Find the values of 3-5 + 2*83 -t-*o-i- 1*175 ; ll*73-10-9ie;
3-375 X 1*6 X 4-8 ; ^* ; and find the product of the results.
29. Find the value of 12 of 3*5 cl 4*375iZ. + 1*83 of *9&4 of -428671
of 4'5df.
30. Find a 5-place decimal nearly=^i^- ^^^- x ilBlilP:.
^ ^ :u22 IGo^ 5 ac. 2 ro.
31. ."Reduce ^ and ^ to decimals ; -65 and -0651 to vulgar fractions ;
and £2 35. Z\d, to the decimal cf £4,
DECIMAL FEACTIONS. 73
32. Find the value of -^85714 of £30 + 6657145;^ + '6 of 't 14286 of
•6£+ 1-S of -42857 i«.
33. Reduce 2| and yfj to decimals ; 2'05 and -206 to vulgar frac-
tions ; and £19 17s. 2jd to the decimal of £5.
34. Multiply 1 cwt. 2 qrs. 3 lbs. by 5 125 ; and divide £3834 05. 6ld,
by 441-75.
35. If an ounce of gold be worth £4-009d, what is the value of a
bar of gold, weighing l-68i lbs. ?
36. Reduce -6 of £1 + -6 of 5«. Zd. + 3-75 of a crown to the decimal
of 168.
37. Find what decimal multiplied by 175 will give the sum of J,
If, 1?, and 3J.
38. Multiply -285 by 4-02 ; divide 2-961 by -007 ; and find the
value of 2-778125 of 6s. Sd.
39. Reduce (^^ of -^\ -j- (— of — i-^ to a simple quantity.
\ 316 -0625/ \ 7 5 6-25/ ^ ^ ''
40. Multiply £2 16«. 10-75(i. by 144-33 ; and divide £9753 14s. S^d.
by 234-5.
41. Find the value of 3-276 of £10; multiply 3-275 by 12-8; and
divide -0625 by -OOOOo.
42. Reduce ^ and ^ to decimals ; 20825 and -3405 to vulgar
fractions; and 2 lbs. 3 oz. to the decimal of a ton.
43. Reduce l-75s. to the decimal of £1 ; and 2*6 of £-877083 to the
decimal of half-a-sovereign.
> 44. Find the value of 3JJ of £3 12s. 6f (?. ; and reduce the result to
the decimal of £35 Os. 3|i.
45. Reduce ?:?-^ + #^^ of ^-^^^ to a simple quantity.
1136 1-6 + 2 629 2 25
46. Find the value of | of 2-625 guineas ; and the difference be-
tween 26-5p. and 70| sq. yds.
47. Find the value of 6-83 of £3-8677083 + 58 of £2 4114583-4 375
of £1-3.
48. Reduce to a decimal, accurate to 5 places,
/i — + — ?_ + &c.^ — —. &« Note VII. !
\6 3x5» 6x5» 7x6' J 239 '
49. Find the sum of £1-15 + 2-0625 guineas + -0078125 of 32s. ; and
reduce the result to the decimal of half-a-sovereign.
60. Reduce to a decimal, accurate to 7 places,
1 +i + J- + — + + &c. See Not^ VYS.,
1 1x2 1x2x3 1x2x3x4
161
74
I
CHAPTER V.
PRACTICB.
68. This is an expeditious method of finding the value of
anj quantity of merchandise, &c. when the valtie of a unit
of amj denomination is given ; as of 456 cwt. at £S IZs. 6d.
per cwt., or of 3 cwt. 3 qrs. 13 oz. at £2 6s. 7^. per lb. &c.
69. Case i. Where the given quantity is expressed in the
same denomination as the unit whose value is given.
Under this head will occur such examples as the follow-
ing : 36 cwt at £S 10s. per cwt, 25 lbs. at £2 ISs. Sd. per lb.,
37 oz. at £5 17 s, 6d. per oz., &c.; where the unit, whose
value is given, is of the same den" as the quantity whose
value is required. It is obviously immaterial what the unit
itself is ; that is to say, the values would be the same either
of 36 cwt at £S lOs. per cwt, or of 36 lbs. at £3 lOs. per lb.,
or of 36 oz. at £S 10s. per oz., or (without specifying any
unit) of 36 articles at £3 lOs. each, or, as it is briefly ex«
pressed, of 36 at £3 lOs.
Ex. 1. Find the value of 36 at £3 lOs,
Here we have, in fact, to multiply £3 lOs. by 36 j let us first then
multiply £3 by 36, or, which amounts to the same, multiply £36 by 3,
and we shall have £108 as the amount of 36 at £3.
£36 Now, instead of multiplying the lOs, by 36, we observe
5 that, since 10*. is £|, we may take 36 times 10*. by
108 taking 36 half pounds, which is the same as taking half
^® of 36 pounds ^£18, which we add to the £108, and
105.
Ans. £126 thus have the whole product of £3 lOs, x 36.
PEACTICB.
75
Kx. 2. Find the value 0/253 at £2 168. Sd.
£253
2
10*.
68, Sd.
506
126 10
84 6
8
Ex.
Ane,
£716 16 8
£2 16s.
£
8.
d.
:.«8.
. 1.
129 at
6
10
3.
157 at
9
5
5.
271 at
8
3
4
7.
289 at 11
1
8
9.
447 at
1
16
8
11.
361 at
9
11
8
Here we find, as before, the value of £2 10*.
X 253 : and then, since 6*. Sd. is £|, dividing
253 by 3, we have £84 6s. Sd., the value of
65. Sd. X 253, which we add to the other two
lines, and thus have the whole product of
£
8.
d
2.
343 at 4
6
8
4.
362 at 7
4
6.
187 at 1
2
6
8.
495 at 12
11
10.
555 at 4
13
4
12.
677 at 2
12
6
£371
5
Ex. 8. Find the value of 371 at £5 17s. 6d.
Here we find the value of £5 IQs. x 371 as
before : then instead of taking, as we might,
65. Sd. as £i, &c., we may take 5s. as J of 10s.,
and so find the value of 5s. x 371, by taking
half the value of 10s. x 371, i.e. half of £185
10s., or £92 15s. : in like manner, we may then
take 2s. 6d. as | of 5s., and find the value of
2s. 6d. X 371 by taking half of £92 15s.
10s.
5s.
28. 6d.
1855
185
10
92
15
46
7
6
Jns. £2179 12 6
Ex. 4. Find the value of 713 at £4 8s. IIM.
£713
4
5s.
8s. 4d.
7J^.
2852
178
5
118
16
8
22
5
7§
Here we find, as in Ex. 2, the value of
£4 8s. 4^. X 713 ; we then take 7^. as | of 5s.,
and so divide by 8 the line £1 78 58.^ which is
the value of 5s. x 713.
Jns. £3171 7 3J
Ex.«9.
£ 8.
d.
£ s.
d.
1. 127 at 3 15
2. 235 at 5 7
6
3. 339 at 4 12
4. 341 at 6 17
6
6. 253 at 7 17
6. 457 at 1 18
6
7. 365 at 11 14
6
8. 573 at 7 15
6
9. 285 at 1 6
6
10. 389 at 8 13
6
11. 492 at 6 18
9
12. 297 at 1 \^
^
76
PEACTIOfi.
9d.
2d.
Ex. 5. Find the value of 89 at 38, llfi.
Here there are no :£* in the given value ; but if
TFO multiply 89 by 3 the result will be in shillings;
then 9d. being | of 3^. we take I of 267^. ; and 2d,
being | of Is. we take J of 89*. ; lastly, |c?. being
895.
3
1
1
e
12
267
66 9
14 10
5
6J
^ of 9d. we take ^ of 665. 9d.
3545. Iii. = £l7 145. l^. Ans.
Ex. 6. Find
£111
65. 8i.
1
37
65. ^d.
37
6s.
1
t
27 15
Sd.
1 7 9
^.
1
u
4 7i
^W5. £103 7
<Ae twZwtf o/ 111 at 185. 7ld,
We might treat this as the last Ex. ; or, to
avoid the final reduction, we may begin at
once by taking 65. Sd. as £J, &c., drawing
a line under the 111, before we divide by 2,
since it is not to bo added in with the other
lines.
4i
^2
£111 X. 9 = £99
6d.
lid.
4
2
18s. Od.
15 6
13 lOj
Ans. £103 7
4i
^2
Otherwise : — As the n* of shillings in the
price is even, we can conveniently change it
into the decimal of £1, viz., £.9, and multi-
plying by .9, we may mentally double the
decimal of the product for shillings ; &c.
Ex. 50. 5.
. 1. 227 at 2
4. 356 at 4
7. 177 at 8
10. 193 at 13
d.
9|
111
61
5.
2. 149 at 3
5. 365 at 5
8. 784 at 9
11. 395 It 14
d.
2|
41
5.
3. 854 at 4
6. 373 at 7
9. 489 at 11
12. 499 at 17
d.
%
111
70. It is often convenibnt to suppose the given valae
increased so as to become an exact number of pounds, or
shillings, &c. for which we may calculate hj common
Mult" ; then, if we find by Practice the value of the part
added to the true value, and subtract it from the other
result, we shall get the required amount.
Thus, in Ex.- 2, supposing the given value to
be £3, we should multiply £3 by 263 ; and then
taking 35. 4d, (the part added to the given
value) as £ J, and subtracting the correspond-
Ans. £716 16 8 ing amount, we have the same result as before.
Similarly, in Ex. 3, we may add 25. 6d. to the given value, making it
£6 ; then multiply £6 by 371, and from the result subtract 25. ed, x 371.
or i of £371.
85. 4cd.
£263
3
759
42
3 4
PRACTICE.
77
Ex. 51.
£ 8, d,
I. 135 at 2 19 3|
3. 273 at 3 18 4%
5. 289 at 8 8|
7. 431 at 5 17 Hi
9. 511 at 7 lOj
11. 271 ate 15 lOf
£ 8.
2. 217 at 4 17
4. 322 at 7 14
6. 373 at 9
8. 397 at 6 15 10
10. 623 at 11
a.
3
91
12. 333 at 5 18 11|
?1. Case n. When the given quantity is not expressed in
the same denomination as the unit whose value is given.
Here wo shall have to find the value of 3 cwt. at £2 13f.6<£
per lb., or of 2 cwt. 3 qrs. 16 lbs. at £3 5s, 7^d, per cwt,,
or per qr., or per lb., &c. In all instances, (like the first of
these,) where the given quantity can be immediately reduced
to the same den'^ as the given unit, we may do this, and shall
then have only an example under Case i. : thus 3 cwt. i=r
336 lbs., and the value of 336 lbs. at £2 ISs. 6d. per Ib-
raay be found as before. So also, if we can reduce any part
of the given quantity to the same den'* as the given unit,
we may find its value by mulf* ; and, for the rem*", we may
take parts of the given unit itself, and proceed as in the
following examples.
Ex. 1. Find the value of 7 cwt 3 qrs, 11 Ihs. at £2 135. Id. per qr.
The given unit being a qr., we reduce 7 cwt. 3 qrs. to 31 qrs., and find
the value of them by multiplying by 31: then to find the value of
11 lbs., we consider that 7 lbs. are J qr., and 4 lbs. are ^ qr. ; so that,
dividing £2 13*. l<f. by 4 and 7, and adding up, we have the value of
7 cwt. 3 qrs. 1 1 lbs.
7 lbs. = I
4 lbs.
_1
Ans, ±83
£ 8.
2 13
d.
Ix
10
31
26 10
10
3
79 12
2 13
13
7
6 >
1
7
raliie
»»
»»
»»
of 30 qrs.
1 qr.
7 lbs.
4 Ihs.
83 6
sj
a\-ps.\\Vo^
a
78 PEACTICE.
Ex. 2. Find the rent of 8a. 3b. IOp. at £l l7s.Sd, per acre.
Here we find the rent of 8a., as in the last example ; and then calca-
late that of 3b. IOf., and add as before.
£ 8. d,
1 17 8x8
8
15 1 4 rent of 8a.
2r.=| 18 10 „ 2R.
iR. =1 9 5 „ IR.
10p.=J 2 41 ^ lOr^
Ans,£l6 11 11| „ 8a. 3b. IOp.
Ex. 3. Find the rent for Smo, Sw, 5d. at £3 13& 6d, per monlA.
Here 1 mo.=4 wks.; and we can take 3wk8.=J of 3 mo. &c, as in
one of the subjoined forms, or 1 wk. 5 da. ^^ of 3 mo. &c, as in the
other.
£ 8. d.
Or thus: 3 13 6
3
£ 8.
3 13
d.
6
3
11
3wk. = J 2 15
4 da. «3|m. 10
1 da.r^i 2
6
6
n
£14 8
9 Ans,
11 6
Iwk. 5da.«i 1 11 6
2 wk. «| m. 1 16 9
£14 8 9
52.
1. 6 cwt. 1 qr. 11 lbs. at £2 17s. 9d, per cwt
2. 3 cwt. 3 qrs. 5 lbs. at £4 145. per cwt.
3. 9 cwt 21 lbs. at £5 Us, l^d. per cwt.
4. 2 cwt 4 lbs. 12 oz. at £3 Is, per cwt
5. 3 qrs. 5 lbs. 9 oz. at £2 I4s. 6d. per lb.
6. 2 qrs. 9 lbs. 13 oz. at I5s, 9d. per lb.
7. 2 qrs. 7 oz. 9 drs. at 18*. 6d, per lb.
8. 2 cwt 2 lbs. 2 oz. 12 drs. at £l 3s. 9d. per lb.
9. 3 cwt 3 qrs. 27 lbs. 15 oz. 12 drs. at £7 per cwt
10. 6 oz. 18 dwts. 20 grs. at 7s, 9d. per oz.
11. 3 lbs. 5 oz. 14 dwts. 12 grs. at 17 s, ed. per oz.
12. 22 yds. 2 ft. 2 in. at 18*. 8d, per yard.
13. 13 yds. 1 ft 7 in. at 9*. 4d. per foot
14. 37a. 1r. 28p. at £2 2«. per acre.
7A lU. 3b. 19p, at £5 18s. 6d. per acre-
I& 2U. 2r. 12 P. at £3 15«. 8«L pet acw»
PRACTICE. 79
17. 5 mo. 3 w. 4 d. at 17«. 6dL per week.
18. 7 mo. 2 w. 5 d. at £2 6s, Id, per month.
19. 9 mo. 1 w. 6 d. at £l 2s. 9d, per week.
20. 6 mo. 3 w. 2 d. at £3 Os, 6cL per month.
72. The method of Practice may be applied, as we hare
said, to any case where the value of any quantity is sought,
that of a unit of any den'* being given. It is not, however,
necessary (as in the foregoing Examples) that this given value
should be the price ©f the unit, &c. ; but, whenever any given
amount is charged for any reason upon the unit, we may
find thus the corresponding amount for the given quantity.
Ex. A bankrupt is able to pay \2s, 6|i?. in the £, and his debts are
£3600 : what was his estate worth ?
3600 ^
-^ This means that, for every £ he owes, he
f ^^^^ can pay only 125. ^d. ; here then we have to
m jQ find the value of 128, 6^ x 3600, which must
have heen the value of his whole estate.
lOs.
25. 6d.
Ans, £2257 10
MlSCELLAl^EOUS EXAMPLES. 53.
1. What mast he paid to 721 labourers for a week's service, at
1 75. 4|d each ?
2. What would be the amount of 137 tons 12 ewt of goods, at the
rate of £2 4s. 10^. per cwt. ?
3. Calculate the amount of a salary of 24418 rupees, valued at
25. 1|(?. each.
4. A bankrupt's debts are £7357f and he is able to pay 12s. 9^. in
the £ ; what are his effects worth ?
5. To how much will a charge of £28 85. 2d. per day amount in 365
days?
6. Lodgings at £5 105. 6d. per month being occupied for 8 mo.
21 days, how much must be paid for them ?
7. What must be given for a gold snufP-box, weighing 5 oz. 9 dwta
20 grs., at the rate of £4 35. dd. per oz. ?
8. What is the dividend on £1710 145. 6d., at 13«. 4J<^. in the £ ?
9. How many acres will supply S3 horses with hay and oa.ts, if each
korse consume annually the produce of 6a. 3b. 26p. ?
10. What is the expense of digging a ditch, of which, tba <i^\Q. ^^"ar
tent is 6756 cubic yards, at the rate of Is. 7|ti. "peT -jatdl
02
80 PEACTICB.
11. A bankrupt owes £2468, and can pay 165. 6d, in the £ ; wliat
are his effects worth ?
12. Find the weight of 1000 pieces of gold coin, each weighing
6 dwt. 7 gr.
13. An officer's pay is 12«. dd. per day ; what is that in a year?
14. A labourer's pay being 2s, 9|rf. a day, what is the whole pay of
23 men for 25 days?
15. If lodgings let at 13*. 6d. per week, how much do they let for
during 273 days?
16. A merchant bought 182 quarters of wheat at £2 Is. Zd. per quar-
ter, and retailed the sama at £2 ISs. 4<?. per quarter ; what was his
gain, and at what per quarter should he have sold it to have gained
exactly 104 guineas?
17. What sum would be required to pay the wages of 377 labourers
for a week, at 2«. 5d. a day each ?
18. If a person's estate be worth £1384 168. per ann., and the land-
tax be assessed at 2s. d^, in the £, what is his net annual income ?
19. An iron bridge consists of 3 arches, the centre one weighing
3046 tons, and the two others 2600 tons each ; what is the cost of the
iron at £6 ISs. 6d. per ton?
20. What will a room cost in painting, at Is. 71^. per square yard,
whose height is 10 ft. 3 in., width 16 ft. 6 in., and length 18 ft. 10 in.?
21. An estate of 134a. 3r. 16p. is rented at £2 I2s. 6d. per acre,
and afterwards the best pasture, consisting of 51a. 2b. 12p., is let at
£3 105. per acre ; what will the first tenant still have to make up of
his rent ?
22. A bankrupt's liabilities are estimated at £3768 17*. 6d. ; what
are his assets, if he can pay 13s. 7^. in the £?
23. What is the joint value of 6 qu. 3^ bu. of wheat at 7«. ^d. per
bushel, and 6 qu. 3J bu. of oats at 4*. 2}d. per bushel ?
24. There were sold throe pieces of land, containing 69|a., 76iA.,
39a. 12p. respectively: the price of the first piece was £12 7^. lOd., of
the second £13 16s. 9^., and of the third £16 8s. 6d. per acre; what was
given for the whole ?
25. What will be the cost of replacing a cistern, to weigh 8 cwt.
2 qrs. 14 lbs., at the rate of £2 Os. 6d. per cwt., if the plumber allows
£1 lis. 6d. per cwt. for the lead of the old one, which weights 6 cwt
I qr. 10 lbs.?
81
CHAPTER VL
PROPORTION.
73. The BaHo of one qaantitj to another is the number
which expresses what fraction the former is of the latter,
and is therefore obtained, as in (48), by dividing the former
by the latter.
Thas the ratio of 108 to 144, or (as it is written) of 108 : 144, is
tH=I* meaning that 108 is J of 144.
The former of the two terms in any ratio is called the
antecedent, and the latter the consequent; and it is plain
from the above, that all ratios are equal which may be made
to have the same antecedent and consequent by striking
common factors out of their two terms.
Thus the ratios of 108 : 144, 36 : 48, 21 : 28, 15 : 20, 3 : 4, &c., are
all equal, since each of them is equivalent to } ; and it will be seen that
the first of each of these pairs of quantities is f of the second. The
learner's attention should now be directed to the Questions on Katio,
p. 183.
74. When two differently expressed ratios are equal, they
are said to form a Proportion, and the four terms composing
them are called proportionals.
Thus, since 15 is f of 20, and 21 is | of 28, and so (as before was
said) the ratio of 15 : 20— the ratio of 21 : 28, these four quantities
form a proportion, which is usually expressed thus, 15 : 20:: 21 : 28,
and read as 15 is to 20 so is 21 to 28, or 15 is to 20 as 21 is to 28; and
here 15 and 21 are the two antecedents, 20 and 28 the two consequents,
of the ratios which form this proportion.
N.B. It should be well noticed that the proportion
15: 20:: 21: 28 expresses that 15 is the same fraction
(proper or improper) of 20 that 21 is of 28.
75. In any proportion, the product of the 1st and 4th
terms = the product of the 2nd and 3rd terms, or, as it is
commonly said, the product of the extremes =tft,c -product qJ
the means.
92 PROPORTION.
Thus in the proportion 16 : 20:: 21 : 28, since the two ratios aiQ
equal, we have |§=f|; and, if we multiply each of these equals by
SO X 28, we get 15 x 28=»20 x 21, or 1st x 4th=2nd x 3rd.
76. Conversely, if the product of any two quantities =
the product of two others, the four are proportionals, the
factors in one product being the extremes, and those in the
other the meansy of the proportion.
Thus, since 6 x 20 = 120 = 8 x 15, if we divide each of these equals
by 6 X 8, 6 x 15, 20 x 8, 20 X 15, respectively, we get
^=^, whence 20 : 8:: 15 : 6
ff = |, whence 20 : 15::8 : 6
f = i§, whence 6 : 8 : : 15 : 20
A =■ A» whence 6 : 15 : :8 : 20
or T^-¥-» whence 15 : 6::20 : 8
or I -ff» whence 8 : 6::20 : 15
or |§ « |, whence 15 : 20::6 : 8
or A =&, whence 8 : 20::6 : 15
in the first set of which proportions it is seen that the terms of one
product, % and 20, are the extremes^ and those of the other product,
8 and 15, the means; and vice versa, in the other set.
77. Hence also it follows, that, if four quantities in any
given order are proportionals, they will also be proportionals
in any other order, in which the same two terms will go
together, either as extremes or means.
Thus, since 6 : 9::lO : 15, it follows by (75) that 6 x 15 = 9 x 10,
and therefore by (76) we have also 6 : 10::9 : 15, 10 : 15::6 : 9, &c.,
in which 6 and 15 still go together, either as extremes or as means. "We
could not have, however, 6 : 15 : : 9 : 10, &c., in which this is not the case.
78. If we have given any three of the four terms of a
proportion, we may by means of them easily find the fourth ;
for since by (75) the 1st x 4th = 2nd x 3rd, we have the
2ndx3rd the 4th = ^"^/ ^^^ the2nd=l?*^
4th ' 1st 3rd
1st X 4th
and the 3rd =
2nd
Ex. Find the numbers which shall form the 1st and 2nd terms,
respectively, of a proportion with the numbers 6, 7, 8.
__ , , 2nd X 3rd 6 X 7 , , ,
Here the lst= — -^ — =-g- = 5i and 5^ : 6::7 : 8j
. ^ , lstx4th 6x8 ^
tho 2nd = 3y^ « -y- = 6f , and 6 : 6f::7 : 8.
PBOPOBTION. 83
Ex. 54. Find numbers which shall form the 1st, 2nd, 3rd, 4th terms,
respectivelj, of a proportion with
1. 2, 3, 4. 2. 3, 4, 5. 3. 4, 5, 6. 4. 5, 6, 7.
5. 2, 5, 7. 6. 4, 5, 8. 7. 2, 7, 9. 8. 5, 7, 7.
79. We have hitherto given instances only of the ratios
of a^f^rac^ quantities, or numbers^ to one another; but we
may similarly obtain the ratios of concrete quantkies.
Thus the ratios of £108 : £144, of 9 cwt. : 12 cwt, of 15 gals.
: 20 gals., of 39 ft. : 52 ft., are (by 48) respectively Jff . ^, |g, ff , each of
which reduces to |; and we say, therefore, that the ratio of £108 : £144
is the same as that of 3 : 4, or f, meaning that £108 is | of £144; and s^
with the other ratios.
Of cours,e, however, the quantities forming such a ratio
must be of the same kind; for, otherwise, one of them could
not be a fraction of the other.
Thus it would be absurd to speak of the ratio of £108 : 144 cwt., or
of 9 cwt : 12 gals., &c
So also, though they be of the same kind, we must besides
reduce them, as in (48), to the same denomination^ before we
can express the one as a fraction of the other, and so find
their ratio.
Thus the ratio of Is, 6d: : 4«. 2d,=^ihQ ratio of 90J. : 50(/.==|§=f
-9 : 5.
N.B. Whatever be the nature of the quantities themselves,
their ratio is always a mere abstract number^ expressing, as
stated in (73), what fraction the one is of the other.
Thus in the last instance the ratio of 90J. : bOd, is, as in (48), the
number |, not §</.; for it has no reference whatever to the fact, that tlie
given quantities were pence, but only to the magnitude of the one with
respect to the other, i. e. to the fact that the one is | of the other; and
it would plainly have been just the same, if the ratio had been that
of £90 : £50, or of 90 cwt. : 50 cwt, &c.
80. So also, when two such ratios are equal, they form a
proportion ; tfius £108 : £144 : : 9 cwt. : 12 cwt. ; only here
we cannot, as in (77), change the order of the terma^ e^^y^s^V
the change be such as still leaves the two x^i^Xo^ po&sVble*
84 PROPORTION.
Thus, it will be true, as before, that £144 : £108 : : 12 cwt. : 9 cwt, or
9 cwt. : 12 cwt.::£l08 : £144, &c.; but we cannot say that £l44 :
12 cwt.:: £108 : 9 cwt., because the two ratios £l44 : 12 cwt. and
igl08 : 9 cwt. are absurd. It would, however, be true, that £144 : £l2
:: 108 cwt. : 9 cwt. &c.
81. For the like reason, we cannot exactly say of such a
proportion, that the product of the extremes = that of the
means ; thus it would be absurd to speak of multiplying £144
by 9 cwt., &c. : if, however, we consider only the numerical
values of the terms, this would be still true*: and, having three
terms of such a proportion given, we may, by means of their
numerical values, find as in (78) the numerical value of the
fourth term, which will be of the same kind and denomination
as the other term of the ratio to wliich it belongs.
Thus to find a fourth proportional to £108, £100, 9 cwt., we have its
100x9
numerical value = .^g = 8 J, which must be 8j- cwt, since it must be
of the same kind and denomination with 9 cwty the other term of the
ratio to which it belongs ; and the proportion will therefore be
£108 : £100 : : 9 cwt. : 8j- cwt.
82. The method above referred to, by which we may find
the fourth proportional to three given quantities, — ^viz. by
multiplying together the 2nd and Srd, and dividing the prO'
duct by the Ist, — is commonly known by the name of the
Rule of Three.
In practical applications of this Rule, the three given
quantities are generally concrete; and a very large class of
Examples are .those, where, the cost of a given quantity of
some article being given, we are required to find, either
what will be the cost of another given quantity, or else what
quantity may be bought for another given cost. For it is
plain that, in any such case, if the first cost be double,
treble, half, &c. of the second cost, the first quantity wiD
be double, treble, half, &c. of the second quantity, and,
generally, the first cost will be the same .fraction of the
second cost that the first quantity is of the second quantity ;
PROPORTION. 85
i. e. the ratio of the two costs will be the same as the ratio
of the two quantities, or the four will be proportionals, so
that we may apply to them the preceding observations.
Ex. 1. If 39 cwt. ofstiffar cost £91, what will be the cost of IS cwt. ?
39 cwt. I 18 cwt. ::£91 : the Ana., whose numerical value we obtain by
18 multiplying £91 by 18, and dividing
728 l>y 39, without considering these as
91 concrete- quantities, and the result 42
39)1638(£42 will be of the same kind as the 3rd
156 term, viz. £■.
78
Ans. £42. 7S^
£z. 2. J[f £42 will buy 18 cwt. of sugar ^ what quantity may be had
for £91 ?
£42 : £91:: 18 cwt
18
yog Here we have multiplied the 2nd by the
91 3rd (the least of the two) for convenience,
r 6^ Tfiis ^^^ ^® result 39 will be of the same kind
*2 i ——-—— as the third term, viz. cwt.
I 7) 273
Ans. 39 cwt.
Ex. 55.
1. If 12 yards of cloth cost £15, what would 8 yards cost at the
same rate?
2. If 46 bu. of wheat cost £16, how many may be bought for £72 ?
3. What will be the cost of 90 gals, of wine, if 495 gals, cost £396 ?
4. How many acres of land may be rented for £65, if the rent of 1 68
acres be £364 ?
5. If 63 loads of straw can be bought for £180, how many may be
had for £100?
6. How much must be given for 25 doz. of wine, at the rate of £176
for 80 doz. ?
83. Since the Answer = ^°^^ ^^, and the value of this
1st
fraction is not altered by striking common factors out cf
its num"^ and den*", we may sometimes simplify the operation
by striking out (before we multiply and divide according
to the Rule) a common factor, either from the 1st and ^vA
terms, or from the 1st and 3rd terms.
8G PEOPOETION.
Ex. 3. Jf 275 reatM of jpaper cost £158 155., what would 990 reams
275 rms. : 090 nns.::£l58 Us, I ^^^^*:^^^^
20 276
127 31765.
^\X\8 X 990 ^ J275. X 90 « 1 14305. = £571 105.
11 .ail5.
Here haying first, for convenience, reduced the 3rd term to shillings,
we have signified the value of the 4th term by a fraction representing
the product of the 2nd and 3rd terms divided by the 1st. Then striking
out 25 from 3175 and 275, -we get 127 and 11 ; then dividing 990 by
the 11, we obtain 127 x 90.
Ex. 4. Jf li: tons of bar 'iron cost £106 ll5. 6d,, how much may he
had for 100 ffuineas?
£106 115. 6d. : 100guin.::14tons : ^^ ^' ^ ^^^^
20 42 *268
2131 4200 sixp.
2
4263 sixp.
200
14x)l5lS^iS(IS^ 2x200
= 40at4-29
^W "" 29
203
29) 400 (13|f tons. Arts.
29
110
J7
23
7. If 386 yards of cloth cost £263, how many may be had for £138?
8. How much cambric may be bought for £46, if 714 yds. cost £86?
9. If 36a. 3b. of land are rented for £84, what should be the rent of
2lA. 3r. 20p. ?
10. If I pay £18 for 7 cwt. 3 qrs. 14 lbs. of sugar, what would be
the cost of 4 cwt. 1 qr. 14 lbs. ?
11. How much oats, at £80 165. for 61 quarters, may be bought for
£62 145.?
12. If 172 cwt. 2 qrs. 18 lbs. of potatoes cost £94 175. 6d,, how much
must be given for 7 cwt. 3 qrs. 11 lbs.?
84. The Principles of Proportion, may, however, be applied
to numberless cases, besides snch as we have been hitherto
considering ; and we must here say a little more of the
general nature of what are called PropcyrtxcnaZ QaantiUes.
PROPORTION. 87
We have already seen what is meant by saying that/bwr
quantities are proportionals ; but it is common also to speak
of two quantities being proportional to each other (or varying
as each other) ; only here the quantities are used generally^
whereas the four quantities^ in the former case, were par*
ticular values of such general quantities.
Thus, for example, we say commonly that the weight of an
article is proportional to, or varies as, the price ; where the
words weight and price are used generally ^ without reference
to any particular weights or priees : but by saying this we
mean, that if we took any two particular weights, and the
two corresponding prices, the four would be proportionals :
and thus, having given any two weights, and one of the
corresponding prices, we might find, by the Rule of Three, the
other price ; or, having given any two prices, and one of
the corresponding weights, we might find the other weight :
and this we have been doing in the preceding examples.
85. But now other quantities, considered generally, may
be similarly proportional to each other ; and to these the
same principles may be applied* Thus, the rent of a house
will vary as the time it is occupied, a workman's wages will
vary as the time he labours, the distance run by a coach will
vary as the rate at which it moves, &c. ; in all which cases,
if we take any two particular values of the first quantity,
and the two corresponding values of the latter, the four
would be proportionals ; so that, if three only were given, we
could apply the Rule of Three, as before, to find the fourth,
86. Sometimes we can only know from philosophical rea-
sons, that two such general quantities are proportional ; as,
for instance, that the length of the shadow cast by a vertical
JTod, at any given hour of the day, varies as the height of the
^od ; that the velocity acquired by a heavy body in falling
Varies as the time of motion from rest, &c. ; but, in moat c^kS^^.'^k
that occur in common practice, it is easy to ap^\y «A. oxv^^ "Oaa
88 PROPORTION.
test of proportionality, as in former examples, viz. by con-
sidering whether, by taking any two particular values of one
quantity, and the corresponding two of the otlier, the four
would be proportional, i.e. whether the 1st of the former set
would be the same fraction of the 2nd, as the 1st of the
latter set would be of the 2nd.
This, perhaps, we may do most simply thus : consider if,
by doubling^ trebling^ &c. any value whatever of the one
quantity, the corresponding value of the other quantity
would also be doubled^ trebled^ &c. ; — in which case the
above test would be satisfied with these four quantities, and
therefore the two given general quantities would be propor-
tional to each other.
87. Hence, when any question is proposed, in which,
having given any two values of one quantity, and one of the
corresponding values of the other, we are required to find the
other of these values, we must first enquire whether the
case be one of Proportion. If so, we may proceed to state
and solve the sum by the Rule of Three. It will be best^
first to set down the 3rd term, which will always be the single
given term, and of the same kind as the answer (being the
antecedent of the ratio in which the answer is the con-
sequent) ; then the other two terms, (which will always be
of the same kind, being two given values of the other
quantity,) will form the other ratio — the antecedent, or let
term, being that value which corresponds to the antecedent
of the second ratio, or 3rd term.
We may now proceed as before — reducing the 1st and
2nd terms to the same den^ — and, if desirable, the 3rd, to
any den^ we please — striking out common factors (if any)
from the 1st and 2nd, or 1st and 3rd — multiplying together !
the 2nd and 3rd, and dividing by the l!?t — ^when the quo-
tient will give the Answer^ in the same den^ as that in
which we have expressed the 3rd term.
PROPORTION'.
89
Uere 5)85m. : 5)130m. ::£l 9
20
17
26
Ex. 1. What is the coach fare for 1 30 miles, if it is £\ 9«. Ad. for 85 miles f
Here it is plain that, if
we double the distance, what-
ever it may be, the corre-
sponding fare will also be
doubled; hence the fare va-
ries as the distance, and we
proceed as before, setting as
the 3rd term the single given
quantity £v 9«. Ad, and, for
the terms of the first ratio,
the pair of given quantities
of the same kind, 85m. and
130m., of which we set 85m.
first, since it is that distance
which corresponds to the 3rd
term.
29
12
352
26
2112
704
17) 9152 (5383!^
85
65
51
Ans. 538^ ȣ2 4 10^.
142
136
6
Ex.2. 77tc rents of a parish amount to £\ 7 50, and a poor-rate is
wanted of £61 I9s, Id, ; what is that inthe£?
£1750 : jC1::£61 19 7 Here it is plain that, if we double the rent,
whatever it may be, the corresponding rate
will also be doubled : so that the rate varies
as the rent. The single term is here the
whole rate, £61 19^. 7d.; the terms of the
first ratio are the two given rents, £1750 and
£1, of which £1750, since it corresponds to
the 3rd term, is set first.
20
• 1239
12
1750) 14875 (8rf.
14000
875 ^
1750 ~"2
Ana, 8^.
This sum in fact amounts merely to one in division; since if £1750
•will supply a rate of £61 19*. 7d., we may obtain that supplied by £l,
by simply dividing this amount by 1750.
Ex. 56.
1. Afield of 18 acres is let for £24 ISs. ed,; what would be the
rent of 42 acres at the same rate?
2. If a servant's wages be £25 a year, what should he receive for 87
days' service ?
3. If the coach fare for 65 miles be £l Is. 8d., how far ought one to
go for £2 18*. 8rf.?
4. If a carding-machine throw off 54 lbs. of wool in 2 hrs. 46 min.
30 sec, in what time will it throw oiF 24 lbs.?
5. How much land may be rented for £70 10s. 6rf., if 5 «jct^% «t^
rented for £4 l3a.Ad,?
90 PEOPORTION.
6. What is the assessment on 20a. , if that on 445a. be £l 4 14«. 9|(f.?
7. If the tax on a rent of £25 is £2 I0«., what will it be on a rent
of £10 9*. 4|<f.?
8. What is the amount of poor-rates to be paid upon £95 lOs. 9^,
when £39 11*. 8c?. is levied upon £791 13*. 4rf.?
9. The expenses of the poor in a parish amount to £110 7*. 6</., and
the whole rent is £2000 ; how much in the £ must be levied to pay it?
10. What is the tax on a house rented at £65 lOs. 6d., if that on
one rented at 25 guineas be £4 Us. 10|(i.?
88. Sometimes we may have two general quantities so de-
pending on each other, that, if we double any value whatever
of the one, the corresponding value of the other, instead of
being doubled, will be halved. Thus, if any given number
of men would do a piece of work in a certain time, it is plain
that double that number would do it in half the time. In
this case the four quantities will still be proportional, but
with the terms of the second ratio in inverted order ; since
the 1st value of the former quantity will be the same fraction
of the 2nd, that the 2nd of the latter quantity is of the 1st.
The two general quantities are here said to be inversely
proportional to each other, whereas in the former examples
they were directly proportional : but the Rule of Three may
still be applied, if we take care to state the sum rightly, viz.
by setting last, as before, the single term, and then setting
as the second term, or consequent of the first ratio, (instead
of, as before, the ^rst, or antecedent,) the corresponding
value of the other two given ones.
Ex. 1. A person completed a journey in 32 days, travelling 8 hrs. 3
day ; how long would he take to do the same, travelling only 6 hrs, a day?
6 hrs. : 8 hrs.::32 days Here the term 8 hrs. corresponds to the
8 term 32 days, and it is plain that if we
6) 256 double the n® of hrs. in each day, the n® of
Ans, 42| days, days required will be only Jialf of what it
was before ; so that the n® of hrs. in a day
varies inversely as the n* of days required. The single or 3rd term is
32 days, and here we put the corresponding term, 8 hrs., second instead
ofjirsf, as in the former cases.
PROPORTION. 91
Or we might reason thus. The whole number of hrs. must be the
same in both cases; and therefore S2 x 8=^6 xAtu., whence we have
the ^«*.=2^«^=42| days.
Ex. 2. If 84 sheep can be grazed in a field for 12 days, how long
might 112 sheep have been grazed in the same field f
Here it is plain that, if we double the
H2: 84^-12* "' ^^ ^^®P' *^®^ ^'^^ ^® ^®P* ^^^ ^"^^
* 12* ^If ^^® *^™® ^^ *^® same field; so that
n2)lTOi'(9clays^n.. Aen- of sheep varies mrer*e/y aj the n-
1008 of aays. The single term is 12 days, and
we set the corresponding term, 84 sheep,
second.
Or thus: 84 sheep for 12 days consume as much as 84 x 12 sheep in
one day; and we have 84 x 12 » 1 12 x Ans. /. Ans.=z^^== 9 days.
Ex. 57.
1. If 100 workmen can do a piece oi work in 12 days, how many can
do the same in 8 days ?
2. If a besieged garrison have 4 months' provisions, at the rate of
18 o«. per man per day, how long would they be able to hold out, if
each man were allowed only 12 oz. per day ?
3. If I borrowed of a friend £300 for 8 months, for how long a
time should I lend him £200 in return ?
4. How many men would perform in 168 days a piece of work,
which 108 men can perform in 266 days ?
6. If a person, travelling 12 hrs. a day, would finish his journey in
3 weeks, how many weeks would he take to do it, if he travelled only
9 hrs. a day at the same rate ?
6. If 47i shilling cakes can be made of a quarter of wheat, what will
be the price of a cake, if 70 are made of the same quantity of flour ?
7. How much land, at 27*. per acre, should be given in exchange for
480 acres, at SSs. per acre ?
8. A besieged fortress has provisions for 3 weeks, at the rate of
14 oz. a day for each man ; at what rate per day must the provision
be distributed, so that the place may hold out 5 weeks ?
89. We must always be assured, as in the preceding
Examples, that the two general quantities concerned in any
case are proportional to one another, either directly or \w-
versely, and so that the question is one vrliida. £«iX\& \xii^«t ^^
92 PROPORTION.
Rules of Proportion. But when satisfied of this, we may
relieve ourselves of some of the care required in stating the
sum, by the following general Rule, which includes both
cases, and is that commonly given as the
RULE OF THREE.
Set last the single term, (viz, that which corresponds to the
Answer,) and the greater or less of the other two terms
second^ according as it is seen that the Answer will be greater
or less than the third term.
Ex. A lent 15 guineas to J?, who repaid the loan at the end of 27
days; what sum, if now lent hy B to A for 20 days, would bean equitable
requital ?
20 days : 27 days :: 31 OS. tt ., . ,
Ans.Sl5s.x27-i-20 ^^^^ ^-^^ sm^^fe, or 3rd, term is 3165.;
= 635. X 27 -r 4 a"d as more Bhould be lent for 20 days
= 425js. = £21 5s. 3d. than for 27, we set the greater of the
other terms in the second place.
Obs. a method of working questions in Proportion, called
the Unitary Method, is by many teachers preferred to the
above Rule. It solves the preceding example as follows :—
Sum B should lend for 27 days = 3155.
„ „ 1 day =3155. X 27
20 days = 3155. xfj
= i of (63 X 27)5. = £21 55. Zd. Ans.
*^* See other examples of the unitary method worked out in Note
Vm. p. 172.
Ex. 58.
1. If 69 lbs. of salt cost 95. l^d., what will be the cost of 15 lbs.?
2. What is the value of sheep per score, if 31 1 sell for £585 l5. 4J<£.?
3. A bankrupt owes £4726 105., and his effects are worth £1181
125. 6d. ; how much will he be able to pay in the £?
4. If 27j bushels of potatoes cost £5 45. 6d., what quantity will cost
£25 145. 7d. ?
5. If 39 cwt. 1 qr. 11 lbs. cost £59 65. Sd., what will 13 cwt. cost at
the same rate ?
PROPORTION. 93
6. What weight of sugar may be bought for £374 8*., when the
cost of 6 cwt. 2 qrs. is £14 145. 8^. ?
7. If the tax on £335 7^. 6d. amount to £68 13a. 9|<?., what is that
inthe£?
8. How many gallons of wine, at the rate of £31 16a. 4d. for 46 gals.,
may be bought for £117 Us. Sd. ?
9. If 17 cwt. 3 qrs. 14 lbs. of tallow cost £38 2s. Sd., how much may
be bought for £5 12^. 6d. at the same rate ?
10. If the sixpenny loaf weighs 3 lbs. when wheat is at 6s. a bushel,
what ought it to weigh when wheat is at 65. Od. a bushel?
11. Suppose there are 12,000,000 sheep fed in this country ; what is
the value of their wool-produce yearly, if 11 sheep produce 25 lbs. of
wool, which is sold at £8 125. per cwt. ?
12. From 3 tons 5 cwt. take 1 ton 16 cwt. 3 qrs. 12 oz., and find the
Value of the remainder at £l 75. 6d. for 1 qr. 27 lbs.
13. If a nobleman's rental be £8050 per annum, and the land-tax be
charged at the rate of £11 55. per £100, what will be his nett income ?
14. If 4J yards of cloth cost £5 145. 4|{f., what would 20 yds. cost?
15. The chain for measuring land is 66 feet long, and divided into
100 links ; what is the length of a wall which measures 2456 links ?
16. The rateable value of a parish amounts to £1250, and a poor-
rate of £27 105. 6d. is to be raised; what will a person have to pay
whose rents are £525 ?
17. A wedge of gold, weighing 14 lbs. 3 oz. 8 dwt., is valued at
£514 45. ; what is the value of an oz. ?
18. A bankrupt has assets to the amount of £1020, and debts to the
amount of £3225 ; what will his creditors receive in the £ ?
19. A bankrupt's effects amounted to £980, which paid his creditors
135. 6^. in the £ ; what did his debts amount to?
20. What is the income corresponding to an income-tax of £ 1 3 25. 6d.,
at the rate of 7 pence in the £ ?
21. A borrowed of JB £175 5s. for 102 days, and afterwards would
return the favour by lending B the sum of £210 6s. ; for how long
should he lend it ?
22. What is the height of a steeple, whose shadow was 148 ft. 4 in.,
at the same time that the shadow of a staff 6 ft. 4 in. long was 5 ft. 3 in.?
23. A coach goes from London to Liverpool, at the rate of 9 miles
an hour, in 24 hours ; in what time would the distance be performed on
the railroad, at the rate of 32 miles an hour ?
24. A besieged town, containing 22400 inhabitants, has provisions
to last 3 weeks ; how niany must be sent away that they may be able to
hpld out 7 we^ks f
94 PROPORTION.
25. If a servant receive £3| for 20 -weeks* service, how manj weeks
ought ho to remain in his place for 12 guineas?
26. If the carriage of 15j cwt. for 60 miles came to 7s. 9f?., how far
ought 3J cwt. to bo carried for the same money ?
27. How much may a person spend in 73 days, if he wishes to lay
by every year 50 guineas out of an income of £450 ?
28. The carriage of a parcel of goods, weighing 1 ton 3 cwt. 2 qrs.,
cost £2 145. ; what will be the charge for 4 other parcels, weighing
each 17 cwt. 3 qrs. 7 lbs. ?
29. If 3| shares in a speculation are worth £27 105., what are 4|
shares worth ?
30. If 1| yard of cotton print cost 25. 6d.f what is the cost of 24i
yards?
31. If 1| cwt. of sugar cost 3| guineas, what must be given for
171 lbs. ?
32. At 35. ^Id. for ^ lbs., what is the price of U| lbs.?
33. If 2J yards of cotton print cost l5. 10^., what is the cost of 13f
yards?
34. If 6§ yards be worth 275. 9|<?., what quantity is worth 185. 2||<?. ?
35. What is the value of f of | of a ship, when f of the whole is
worth £525?
36. If 6336 stones of 3J fl. length complete a certain quantity of wall,
how many similar stones of 2| ft. length will raise a like quantity ?
37. If a ball falling from rest acquire a velocity of 115J ft. in 3|
seconds, at what rate will it be moving at the end of the firet second,
and at the end of 4| seconds ?
38. What will 3 cwt. 1 lb. Ij oz. of merchandise cost, if the cost of
13f tons be 500 guineas?
39. If 4f oz. Av. cost 8|j5., what will 8^ lbs. cost?
40. If 3^ of I of 2i of 40 lbs. of beef cost 1^<?., how many lbs. ma^
be bought at the same rate for 6s. 7^. ?
I
90. Suppose it were asked, * If 9 men can reap 30 acres
of wheat in 10 days of 6 hours each, how many men would
reap 40 acres in the same time ?* This would be an instance
T)f common Direct Proportion, and we ghould have
30a. : 40a. : : 9 men ! |^ x 9 = 12 men.
But now suppose that, instead of *m the same time,* the
gnestion had said, *in 12 days of the same length.* Here it
PROPORTION. 95
is plain that, after finding, as above, the n** of men, 12, who
would reap 40a. in 10 days, we must still have another Pro-
portion, to find the b9 who will reap the same n® of acres in
12 days J thus (the case being here one of Inverse Proportion),
12 days : 10 days :: 12 men : — x 12 men=10 men.
Once more, suppose that, instead of * 12 days of the same
length,^ the question had said, * 12 days of 7^ hrs. each.'
Here, after having found, as above, the n° of men, 10, who
will reap the 40a. in 12 days of 6 hrs. each, we must still
have a third Proportion, to find the n<> who will reap the
same n^ of acres in the same n^ of days of 7^ hrs. each ;
thus (the case being here also one of Inverse Proportion),
7^ hrs. : 6 hrs. 1 1 10 men : =7 X 10 men = 8 men.
'^
91. Now the above is an instance of Compound Proportion,
whereas the preceding Examples were all instances of Simple
Proportion ; the difference between questions in Simple and
Compound Proportion being, that, in the former, we have
one general quantity proportional to another; whereas, in
the latter, we have one general quantity proportional to
each of several others, taken separately, i. e. supposing that,
while we take the two different values of any one of them,
the others meanwhile retain the same fixed values.
Thus, in the above Proportions, the n<> of men is proportional,
in the 1st, to the n« oi acres (directly) when the n*» of days continues
the same^ and the n® of hours in each day the same —
in the 2nd, to the u" of days (inversely) when the n® of acres contin'?es
the same, and the n® of hours the same —
in the Srd, to the n« of hours (inversely) when the n® of acre* continues
the samcj and the n® of days the same.
92. We have seen that, in cases of Simple Proportion,
when a single value of one general quantity is given cor-
responding to one given value of the other, "wa TXi^-^ ^xv^ ^-^^
h2
96 PROPORTION.
corresponding to another given value of the other by the
Rule of Three. In like manner, in cases of Compound
Proportion, when a single value of the first quantity is
given, corresponding to one given set of values of the
other quantities, we may find that corresponding to another
given set of them, either, as above, by successive Propor-
tions^ or by what is called the Double Rule of Three,
which arises from the following consideration. Taking
the numerical value of the 1st result in its original form,
40 10 X 40
-- - X 9, we have that of the 2nd, ^o qa ^ ^> ^^^ ^^ *^®
3rd, ^ , =r^ ^^ X 9, which would, of course, reduce itself
7-jr X 1^ X Ov/
to the final answer, 8, i. e. 8 men : but now this is the same
result as we should get, if we made only one statement,
in which we set down the single term, 9 men, as usual, last,
and, for the 1st and 2nd terms, the products, respectively, of
the numerical values of the 1st and 2nd terms of the three
Proportions.
The same will be true in other cases. It is best to set down,
one under another, the num. values of the^r*^ ratios of these
Proportions, observing to state them by considering each
general quantity separately, with reference to that quantity
whose single value forms the 3rd term; and then we may
multiply these together, (striking out, as before, common
factors from the 1st and 2nd, or 1st and 3rd,) and, finally,
multiply together the 2nd and 3rd terms of the resulting
compound statement, and divide by the first.
Ex. If 5 compositors set up a work of 6 sheets in 8 days, in what
time will 6 compositors set up a work of sheets ?
Here 8 days is the single term, to be set last : now, if we doubled the
n° of men (supposing the same n' of sheets), the n*» of days would be
halved ; hence the n* of days varies inversely as the n° of men, and tJie
corresponding first ratio will be 6 men : 5 men. Again, if we doubled
the n« of sheets (supposing the same n° of men), the n* of days would be
doubled; hence the n» of days varies directly as the n« of sheets, and the
I
PROPORTION. 97
corresponding first ratio will be 6 sheets : 9 sheets ; we have^ therefore,
setting down the numerical values of these ratios),
6 . o I ft V fi X 9
•51 8x5
^::8da. : -^—
da.
juid now striking out 4 from the dividend, and 2x2 from the divisor,
we have
2
^x5x9
^ « =2 X 5 = 10 days. Ana.
3 3
Ex. 59.
1. If 15 pecks of wheat serve 9 persons for 22 days, how long will
20 pecks serve 6 persons ?
2. If £33 55. pay 15 labourers for 18 days, how many labourers
will £79 165. pay for 24 days?
3. If 27 men can dig 2\ acres in 2 days, how many men can dig 2
acres in 3 days ?
4. If 7 horses be kept 20 days for £12, how many may be kept 14
days for £18 ?
5. If 9 persons spend £147 in 6 months, how many will £130 135,
^d. last for 4 months ?
6. If 6 horses consume 375 lbs. of oats in 8 days, what quantity will
4 horses consume in 10 days?
7. How much paper is required for 5000 copies of a book of 12 J
sheets, if 66 reams are required for 3000 copies of a book of 1 1 sheets ?
8. If 8 men earn £9 wages for 5 days' work, how much would 36
men earn for 24 days* work at the same rate ?
9. If £100 will pay the expenses of 5 persons for 22 wks. 6 da., how
long would 12 persons be supported by £150 under similar circum-
stances?
10. If 7 men earn £9 IO5. Qd, in' 10| days, what sum will 28 men
earn in 31|days?
11. If the wages of 25 men amount to £115 in 16 days, how many
men must work 24 days to receive £155 55., the daily wages of the latter
being one-half those of the former ?
12. If 21 men mow 72 acres of grass in 5 days, how many must be
employed to mow 460a. 3b. 8p. in 6 days ?
13. If 9 persons spend £120 in 8 months, how much will serve 26
persons for 12 months?
14. If 12 horses in 4| days plough 10| acres, how many horses TiirQ\^4.
plough 35 acres in 20 days ?
98 PROPORTION.
15. If a 3 lb. loaf costs Td, when wheat is at 625. 6^. per quarter,
what should be the price of wheat when a 2 lb. loaf costs b\i. ?
16. If a man travels 65 miles in 3 days, by walking *!\ hours a day.
in how many days will he travel 156 miles by walking 8 hours a day ?
17. What will be the wages of 15 men for 10 months, when 9 men
receive £261 15s. for 8 months?
18. If 3 persons are boarded 5 weeks for £17 105., how long should
14 persons be boarded for 60 guineas?
19. How far should 80 cwt. be carried for £29, if 30 cwt. be carried
17 miles for £5 85. ^d. ?
20. If 6 men can reap 34 acres of com in 5 days, how many men will
be required to reap 95a. 32p. in 10| days?
21. If 40 bushels of corn serve 12 horses 37 days, how many days
would 195 bushels serve 9 horses ?
22. A person completes a journey of 160 miles in 3 days, travelling
11 hours a day; in how many days would he complete 1000 miles,
going 15 hourls a day at the same rate ?
23. If 3 men can reap 7 acres of wheat in 2 days, how long will it
take 8 men to reap 20 acres at the same rate ?
24. If a ton of turnips will last 25 sheep for a fortnight, how much
will be required to supply 40 sheep during the months of January and
February in Leap-year ? '
25. If 6 men can dig a trench, 220 yards long, in 2|days, by working
8 hours a day, how many will dig a trench, 187 yards long, in 4J days,
working 6 hours a day ?
26. If 12 men build 24 rods of wall in 30 days, working 8 hours a
day, how many hours a day must 18 men work to build 64 rods in 40
days?
27. If 8 men can plough 84 acres in 12 days of 8 J hours each, how
many acres can be ploughed by 20 men in 11 days of 7| hours each ?
28. If 8 men can dig a trench 100 ft. long, 3 ft. broad, and 4 ft. 6 in.
deep in 9 hours, how many will be required to dig a trench 80 ft. long,
5 ft. broad, and 2 ft. deep in 5| hours ?
29. If 7 men can erect a certain piece of wall in 20f days of 9| hours
each, how long would it take 3 men to do 2| of the same work, reckoning
10| hours to the day ?
30. If 20 men can excavate 185 cubic yards of earth in 9 hours, how
many men could do half the work in a fifth of the time?
99
CHAPTER VII.
MISCELLANEOUS RULES.
93. Interest is the consideration paid for the use of money.
The Rate of Interest is the sum paid for the use of a certain
sum, generally £100, for a certain time, generally one year :
thus, if £5 is paid for the use of £100 for one year, the in-
terest is said to be at the rate of 5 per cent.
The sum originally lent is called the Principal; and the
principal, together with its interest for any time, is called
the Amount for that time.
When interest is only taken for the original principal, it
is called Simple Interest; but, when at the end of any
stated period, as a year, the interest accruing is added to
the previous principal, and interest reckoned upon this sum,
taken as the principal, for the next year, it is called Com-
pound Interest.
94. To find the Simple Interest on a given sum for a
given time at a given rate per cent, per annum,
KuLE. Multiply the principal by the number of years,
and by the rate of interest per cent., and divide the result
by 100 ; the quotient will be the interest required.
Ex. 1. Find the Simple Interest on £725 for 3 years at 5 per cent,
per annum,
£725 For the Int. will be the same, whether we
suppose the Principal, £725, repeated three times
2175 in three successive years, or three timesi in one
^ and the same year ; that is, the Int. on £725
108'75 for three years is the same as the Int. on
20 £2175 for one year: and this we find, accord-
Ans, £108 15«. 15*00 ing to the above definition of Int., by dividing
by 100, to see how many Cents there are in the sum, and then taking 5
for each, i, e, multiplying by 5 ; or, which is the same thing, hal xssaxsk
convenient in practice, we first multiply by 5, au^ \X\t^ ^\n\^^\>1 \Q>^
100 INTEREST.
Ex. 2. Find die Simple InUrest on £212 10«. Ad. for 2| yrs. at 2| per
Here the rem', after dividing by 100, is
100 200 40
and, the Int. being £U 12«. 2|i</., we have the
whole amount £227 2«. 6jJ<i. But it is gene-
rally best to represent the whole procedure first
symbolically, in order to ascertain whether the
calculation may be simplified; thus we have
£212 IPs. 4(/. x2!x2| _ £212 10^. 4d.^U
100 160
so that ^ of the given principal will be the
interest
2'52i
Ex 60. Find at Simple Interest,
1. Interest on £500 for 6 yrs. at 5 per cent
2. Interest on £375 for 3 yrs. at 4 per cent
3. Amount of £11 25 for 4 yrs. at 3 per cent.
4. Amount of £2275 for 3| yrs. at 5 per cent.
6. Interest on £347 165. Sd. for 15 yrs. at 4f per cent
6. Amount of £2000 for 12i yrs. at 3J per cent
7. Amount of £575 for 8J yrs. at ^ per cent.
8. Interest on £325 10*. for 4 yrs. at 5| per cent.
9. Interest on £500 13«. 4d, for 22 yrs. at 25 per cent
10. Interest on £ 1 50 for 3^ yrs. at 4 per cent
cent, per ann,
£212 10
4
_2|
fori
fori
425
106 6
53 2
8
2
7
f.T I
584 8
5
2J
1168 16
292 4
10
n
14-61 1
20
12-21
12
oj
If parts of a year be given, they may be expressed as a
fraction of a year.
Thus the Int. for 2 yrs. 3 ma, at any given rate, would be the same
as that for 2 J yrs. at the same rate.
But, in practice, more accuracy is generally required;
and we must express the given parts of a year in days, and
then, finding first the Int. for one year, we may find by a
proportion the Int. for the given portion of a year.
Ex. 3. Find the InU on £Z25from March 1, 1871, to May 31, 1874,
at 4 per cent, per ann.
INTEREST.
101
When interest is thas required from one date to another, the day of
the first date is to be left out, because it is not until the day following
that one day's interest will have accrued. Accordingly, we have here
the whole time=3yrs. 91 da.
Now, the int for 1 year is (:e325 x 4) -r 100 =£13; and for 91 days we
have by Proportion —
365 da. : 91 da. :: £13 : £3 4s. 9|M
Int. for3yrs.=:£l3x3= 39
Ans, The whole int. is £42 4
9S^
If the rate of Interest be given in paris of a £, they may be
expressed as & fraction of a £, and the sum treated as before ;
or we may work for them by the method of Practice.
Ex. 4. Find the Int on £500 for 4 yr*., at £5 7*. 6d. per cent
£600x4j^^g^21|=£l07 10.. Ans.
100 *
Ex. 5. Find the Int. on £307 15s. 6d. for 156 days, at £4 145. ed,
per cent.
Here it will be best to work throughout by decimals, and to extend
them only to so many places as will insure the accuracy of the final
result to two or three decimals of a penny. Also we may employ the
common expedient of doubling the rate and the divisor 365 ; thus we
have the int. = £3-07775 x 9^ x 156-^730.
£3 07776=* Principal -^ 100
^20
3
30
10
10000
20)27*69975
1-384987 5
290847375 x 166
1454236875
1745084250
730 )4537-2190500
Ans. £6-2153685 = £6 45. SeSSd.
4'53721905 We may here exemplify Hunter's useful
artifice in division by 730. It is as follows: —
Write first the 1000th part of the given divi-
dend ; divide it by 3, 10, and 10 successively,
and add the quotients ; the sum diminished
by its 10000th part will be an extremely
near approximation to the true axi^^^t. (^^aa
1-61240635
•16124063
•01612406
6-2169901
•0006216
£6-2163686
Hunter's Exam. Questions on CoUnsds Algebra^ Ai^^. 1.^
102 INTEREST.
Ex. 61a Simple Interest
1. Find the amt. of £500 from March 1 to Jan. 10, at 4| per cent.
2. Find the amt, of £7500 from May 5 to Oct. 27, at 3| per cent.
3. Find the amt. of £l 1 58 1 7s. Gd, for 1 yr. 1 1 5 d., at £2 1 0*. per cent.
4. Find the int. on £260 125. 6d, from March 26, 1870, to Oct. 31,
1872, at 3 per cent.
5. Find the int. on £3996 1 5s. for 4yrs. 225 d. at £2 13s. 4d. per cent
6. Find the int. on £2755 15«. for 3yrs. 1 10 d. at £3 2s. Od, per cent
95. To find the Compound Interest on a given sum^for a
given time, at a given rate per cent, per ann.
EuLE. At the end of each year add the Interest for that
year to the Principal at the beginning of it, and this will be
the Principal for the next year ; and so on, till we have
found the final Principal, or whole Amount, See Note IX.
Ex. Find the Compound Interest on £750 for 3 yrs,, at 4 per cent
per ann, ; and also at 2i per cent, per ann.
£760 First Principal.
3^=s 30*00 Int. in 1st year.
780-00 Second Principal.
5^= 31-20 Int. in 2nd year.
811-20 Third Principal.
^= 32-448 Int in 3Td year.
£8 43-648 -760 = £93 125. U^. 1st Ans,
2 , £760 First Principal.
— ^ = T7i= 18*75 Int. in 1st year.
100 *°
768-75 Second Principal.
5^0 = 19-21875 Int. in 2nd year.
787-96875 Third Principal.
^=: 19-69921876 Int. in 3rd year.
807*66796875 -750 = £67 135. 4{^d. 2nd Ans.
Ex. 62.
1. Find the amt of £95 1 65. 8f/., for 2yrs., at 2j per cent at comp. int
2. Find the amt of £50, for 3 yrs., at 5 per cent at comp. int
3. Find the difference between the simple and compound interest on
£41 135. Ad., for 2 years, at 5 per cent
4. Find the difference hetween the simple and compound interest on
£365 45. %}d.y for 3 years, at 4 per cent.
5, Find the comp. int on £225, for 3 years, at 3| per cent
64 Find the comp. int on £300, for a yeaia, ©X 5.\"^t cawt.
INTEREST. 103
96. There Bxefour things to be considered in all questions
of Interest— the Principal^ the Rate of Interest^ the Ti/wc,
and the Total Interest^ (the Amount being only the sum of
the first and last of these) ; and, if anj three of these be
given, we are able to obtain the fourth. Hitherto we have
only considered the case which most commonly occurs in
practice, viz. that in which the Principal, Rate, and Time
are given to find the Interest, (or the Amount) ; we shall
now give an Example of each of the other three cases which
may arise in Simple Interest — those in Compound Interest
being more difficult, and of less frequent occurrence.
L When the Principal, Interest {or Amount), and Bate
are given to find the Time.
Ex. In what time will £91 13«. 4d amount to £105 65. 0^, at 4j
per cent per ann. ?
Subtracting the principal from the amount, we have here given the
interest^£\Z 12«. %^.\ now in one year £91 13«. 4rf. produces, attho
given rate, 4?iyii=^iL^; we have, therefore,
^^iiiiH : £13 12«. ^d. :: l year,
or, £1) X 17 : £13 12«. 8|<f. x 48 :: l year.
12
163 12 6
4
11 )654|
17 ) 59|
3| years. An&^
11. When tJie Rate, Time, and Interest (or Amount) are
given to find the Principal.
Ex. What snm of money, put out to interest for 4 yrs. at 3| per cent.,
will amount to £259 78, ?
At the given rate for the given time the interest of £100 would be
£3jx 4»£14, and therefore its amount £ll4 ; we have, therefore,
£114 : £259 7s.::£lOO : the Ans.f
which we obtain in the usual manner, ^ £227 \03«
104 INTEREST.
III. When the Principal^ Time, and Interest (or Amount)
are given to find the Rate.
Ex. 1. At what rate per cent, will £142 10«. amount to £163 13«. 1 1 \d.
in 4i years?
The interest of £142 10«. is £21 3«. \\\d. in 4| years,
/. for 1 year it is 20349c/. -5- 1 7 « 1197rf. ;
and £142 10«. being »34200d, we have
34200c/. : £100 :: 1197c/.
or, 38c/. : £1 :: 133c/. : £3|.
^n£. 3| per cent per ann.
Ex. 2. At what rate per cent, per annum will £5 amount to
5 guineas in 219 days?
In 219 da., or f of a year, the int. of £5 is 5«.
/. in I year it is 5«. -^| = 8|fc
£5: £100 :: 8J«. : £8 J.
Arts, 8| per cent, per ann.
Ex. 63. Simple Interest
1. At what rate will the int. on £l02 10«. amount to £12 13«. 8^
in 2i years ?
2. What sum will amount to £45 0*. 9|c/. in i year, at Gj per cent. ?
3. In what time will the int. on £498 16«. 8c/. amount to £10 9«. 3^c/.,
at 6| per cent?
4. At what rate per cent, will the int. on £200, for 146 days, amount
to £4 16s.?
5. In what time will £732 lis. 10c/. amount to £1709 Is. 7J</., at
h\ per cent. ?
6. What sum must be put out to interest at 4§ per cent, to become
£49 Os. b\d. in 5\ years ?
7. At what rate will the int on £4127 10s. amount to £92 17*. A^.
. in a year ?
8. What principal will produce £121 15s. bd. in 2 yrs. 1 mo., interest
at 5| per cent. ?
9. In what time will £419 amount to £486 4s. 3ic/., at 4| per cent ?
10. At what rate will £220 12s. 6c/. become £240 4s. 8|</. in 3J yrs. ?
11. What principal in 3 years 73 days will become £10 Is. lOjc/.,
interest at 6| per cent. ?
12. In what time will the interest on £812 lOs. 10c/. amount to
£771 18s. 3lc/., at 4| per cent ?
DISCOUNT. 105
97. Discount is the sum allowed for the payment of
money before it is due.
Thus, if A has to pay to B £525 at the end of a i/ear, and the rate
of interest is 5 per cent., he might arrange to discharge his dcht by
paying him now £500, because this sum put out to interest would
amount to £525 at the year's end. In this case, therefore, £25 would
be the discount which B would allow to A^ for paying him the debt at
the present time.
The present value of a sum, due at some future time, is,
therefore, the sum left, when the discount for that time is
deducted, (as £500 in the above instance); and may be
defined to be that sum which, put out at interest for the
time in question, would amount to the sum due at the end
of the time ; and the discount is the difference between the
whole sum and its present value, or the interest upon the
present value.
D8. The most common form in which Discount occurs is
in the prepayment of Bills or Notes of Hand, which are both
documents (but differing somewhat in form and character)
by which a person engages liimself to pay a certain sum, at
a certain future time, both named therein. If the credit of
the party promising payment, or of the party holding the bill,
be considered satisfactory, a banker will discount it, that is,
will pay its present value at once, deducting from the whole
amount the discount upon it for the time that must elapse
before it will become due.
99. In practice, however, it is usual to charge as dis-
count the interest on the future debt itself; by which means
the present value obtained is evidently less than it should
equitably be.
Thus, if a banker discounted at 5 per cent, a bill for £525, due nt
a year's end, he would not calculate what sum (viz. £500) at interest
would produce £525 at the year's end, and so deduct the interest
(viz. £25) for this sum as discount ; but he would calculate the interest
on the debt, £525, itself (viz. £26 5«.), and, deducting this^ would pay only
£498 1 5s. to the holder of the bill as its present value. B^ x\i\^\cv^'d;\N.%^
since £498 15«., with its own interest, would not txiiouxit xo £^^^ Vcv ^
106 DISCOUNT.
year, the holder is a loser, and the banker gains, as we have seen, the
difference of £500 and £498 15^., viz. £\ 5«., by the transaction — being,
in fact, the interest upon the true discount
In practice, therefore, questions in discount are deduced
merely to questions in Simple Interest ; but we shall, here
and throughout, give examples in the more correct rule,
unless the contrary be expressed.
N.B. In Great Britain and Ireland 3 days, called Days of Grace,
are always allowed, after the time that a bill is nominally due, before it
is legally due. Thus, if a bill of £250 were drawn on July 10» at 3
months, it would be nominally due on Oct. 10, but legally oa Oct* 13 ;
and, if a banker were to discount it on Aug. 20, he would reckon forward
to Oct. 13, (the last of these days inclusive,) and, finding the interval to
be 54 days, he would reckon the interest on £250 for that time, and,
deducting it as discount, would pay the difference as the present value
of the bill.
It may be noticed, also, that, if a bill would fall nominally due on the
29th, 30th, or 31st of February, or on the 31st of any monUi which has
only 30 days, it is considered to be nominally due on the last day of the
month, and therefore legally on the 3rd of the following month : and, if
any fall legally due on Sunday, they are paid in Great Britain on the
Saturday, but in Ireland on the Monday.
Ex. 1. What is the discount on £396 17 s. 5}d., due at 9 months, at
4 per cent. ?
This example falls under (96), Case 11, in Simple Interest; since,
therefore, £lOO produces in 9 months, at 4 per cent., £3, we have £100,
the present value of £103, due at the end of 9 months ; and thus we get
the proportion,
£103 : £396 175. 5j£f.::£lOO,
which, being solved as usual, gives us the present value £385 6«. 3rf., and
therefore the discount, £11 lis. 2^.
Ex. 2. What would a banker gain by discounting on Sept. 21 a bill
of £318 3*., dated July 31, at 4 months, at 5 per cent. ?
This bill will be nominally due on Nov. 30, and legally on Dec. 3 ;
and, reckoning from Sept 21 to Dec. 3, (the last inclusive), we have
73 days. We shall find the interest on £318 Ss. for 73 days, in the
usual manner, to be £3 3^. 7^. ; and the present value of it, Le. that
principal, which at 5 per cent, would become £318 Ss. in 73 days, we
shall find, as in Ex. 1, to be £315, and therefore we have the discounts*
j^S 3s, i BO that the banker gains upon the whole *l^fL
INSUEANCB. 107
Ex. 64.
1. Find the present value of £284, duo at the end of 2 years, at 3 J
per cent, per annum.
2. What is the present value of £850, due at the end of 3 years, at
3| per cent. ?
3. Find the discount on £1336 lis. Zd., due at the end of 3| years,
at 6 per cent.
4. Required the present value of £151 17*. 6d., due at the end of 4
years, at 6| per cent.
5. What is the discount on £88 2^. 5d.t due at the end of 5 months,
at 4J per cent. ?
6. Find the discount on £210 12^. Itf., due at the end of 3| years, at
ij per cent
7. Find the present value of £598 95. dd.^ due at the end of 1 year
115 days, at 2| per cent.
Find the true discount upon the following bills —
Drawn. Discounted.
March 6, at 7 months Sept. 15, at 5 per cent.
Sept. 12, at 5 months Jan. 13, at 4 per cent.
Feb. 29, at 3 months April 27, at 3J per cent.
March 17, at 3 months May 31, at 6 per cent.
Aug. 5, at 5 months Dec. 6, at 85 per cent.
May 31, at 4 months Sept. 3, at 5 per cent.
Dec. 25, at 2 months Feb. 8, at 6 per cent.
See Note X.
£ s.
d.
8.
419 12
1
9.
457 18
10.
637 5
2
11.
755 5
9
12.
1006 15
6
13.
1337 14
6
14.
1846 5
2
100. There are other cases of common occmrence in
which a rate per cent is charged.
Insurance is a per centage paid for securing property from
fire, &c. The charge is regulated by the nature of the pro-
perty insured, and the hazard to which it is exposed, as laid
down in the Tables of the different Insurance Companies,
The whole annual payment is called the Premium, and the
leg^l document by which the Insurer is secured from loss
is called the Policy of Insurance,
Life Insurance is a per centage paid for securing the pay-
ment of a sum of money upon the death of a person. The
charge is regulated by the age and healthiness of the person
whose life is assured, at the time the Policy YiSb'^ ^ix«X» \si>sA"^
108 INSURANCE, ETC.
out, as laid down in the Tables ; and, being thus settled, it
is reckoned per cent, upon the whole sura secured — the
whole annual payment being called, as before, the Premium
upon the Policy of Assurance,
In each of the ahove cases the Premium, Hkc Interest, must be
renewed ewery year, while the Policy is in force ; but the following charges
are, from their nature, paid only once.
Insurance from sea risk is a per centage charged upon
the value of a cargo, just as in Fire Insurance.
Commission is a per centage paid to an agent for buying
or selling goods.
Brokerage is a smaller per centage of the same nature,
paid usually for transacting money concerns.
101. It is usual with tradesmen to allow (what is called)
a discount of 5 per cent, for ready-money payments upon
goods purchased, or, (since 5 per cent, is the same as 1 in
20), to allow a shilling in the potmd upon the account to be
paid : thus, for ready-money payment of an account of
£7 135. Q>d,^ most tradesmen would allow Is. 6c?., (7^. for
the £7, and 6d, for the 10*.,) and would be content there-
fore to receive as full payment £1 65. This, however,
differs from the discount of which we have before been
speaking, since it takes no account of the timcy at which the
debt would otherwise be paid ; but is merely an arrange-
ment to secure to the seller the convenience of a ready-
money payment, by giving to the buyer a corresponding
advantage.
Ex. 1. What is the sum to be Ex. 2. What is the premium
paid fur insuring a vessel and cargo,
worth £2225, at 3j per cent.?
£2225
3^
I 6675
for \ I 556 5
72-31
20
6-25
12
-///A £72 6s. 3d. SrOO
upon a policy of £375 upon a life
of 28, the rate being £2 8«. 7rf.
per cent, for that age?
Here £375 = 3| of £100; and
the premium is 3| of
£2 8 7
3
4)7 5
1 16 .51
£9 2 2| Ana,
STOCKS. 109
Ex. 3. What sum should be insured at 4 per cent., on goods worth
£725, that the owner may receive, in case of loss, the value both of
goods and premium ?
Here, if £lOO were insured, it would cover goods to the amount of
£96, together with the premium £4 ; hence we have the proportion
£96 : £735:: £100,
whence we get, as usual, the Ans.==£765 I2s. 6<L
Ex. 65.
1. What would be the ready-money payment of an amount of
£27 13«. 6c?., discount being allowed at 5 per cent. ?
2. What would be the expense of insuring a vessel and cargo, w hose
value is £2516 10«., at 3| per cent. ?
3. What is the premium on a policy of assurance for £2286 135. 4d,,
upon the life of a person aged 42, at the rate of £3 lOs. per cent, for that
age?
4. At 4| per cent., for what sum should goods be insured, which are
worth £427 15*. 3</., in order that, in case of loss, the owner may recover
their value, together with the premium paid ?
5. What would be the cash payment of an account of £27 175. 5</., at
5 per cent. ?
6. What is the brokerage upon a money transaction of £273 15«., at
Ss. Ad. per cent ?
7. For what sum should a cargo, worth £5263, be insured, at 7| per
cent, so that the owner may recover, in case of loss, the value both of
cargo and premium ?
8. What is the commission upon £713 6*. 8</., at 2| per cent ?
9. What is the premium of insurance upon £3208 lis. Id, at £2 12«.
per cent ?
10. What is the premium on a policy of insurance for £1237 lOa^
upon a life of 21 years, at the rate of £2 2*. 4d, per cent for that age ?
1 1. What is the brokerage on £768 2s, 6d., at 3«. 4d, per cent?
12. For what sum should goods, worth £4384 0*. Sd., be insured at '
£2 6s, Sd, per cent., that the owner may recover, in case of loss, the
value of both goods and premium ?
102. Stock is the name given to Money, lent to some
Trading Company, or, more commonly, to our own or some
foreign Government, at some given rate of Interest, which
is settled at the time the Money is first lent, according to tlv^
circumstances then existing.
I
110 STOCKS.
Thus, if Government were to borrow to the amount of ^500,000 at
4 per cent., and A had lent £100 of this sum, A would be said to have
£100, 4 per cent stocky and would receive a document entitling him to
receive the Interest (viz. £4) upon this stock from year to year, until
Government chose to repay the Principal, and put an end to the debt
The source from which the Interest is paid is called the
'Public Funds,' being, however, only an imaginary Property,
representing the credit of the Country itself, which is pledged
to the payment of the debts contracted by its Government ;
the Interest is paid half-yearly, and the document, entitling
the possessor to receive it, may be sold, and transferred from
one party to another, just as any other kind of property.
If money would always bring the same amount of Interest,
the average price of £100 stock would be always the same,
(viz. £100, the price first given for it) — we say the average
price, because even then the price would evidently be some-
what less immediately after the payment of a dividend than
it would be immediately before it. But not only does this
cause affect the price of Stocks, but the continual fluctuations
in the value of Money, arising from commercial or political
changes or expectations abroad and at home, are constantly
disturbing it, even two or three times in the same day,
according to the news which reach ua. The price of stock,
then, will rise or fall according as it seems most likely that
Money would fetch elsewhere a higher or a lower rate of
Interest, i. e. would be more scarce, and in demand, as in
prospect of war, or of active speculation, or be lying upon
hand and plentiful, as when trade is looking dull, and there
are no means of employing capital.
Thus, if at the time A wished to sell his stock, money was elsewhere
making 5 per cent., it is plain that no one would give him £100 for the
right to receive only 4 j but since £80 of common or sterling money
(as it is called) would now bring £4 interest, he would be able to sell
his £100 stock for £80; and the 4 per cents, would be said to be
selling at 80.
With this explanation, the mode of treating questions on
Slocks will be easily seen from the following Examples.
STOCKS. Ill
Ex. 1. If £3500 be invested in the 3|'per cents, at 98, what is the
annual income thence derived ?
Here — — ---n« o{ cents, purchased, for each of which £3| are paid as
interest : hence the whole incomes-^ x 3^=£l25.
98 *
Ex. 2. The 3| per cents, are at 99| ; how much money must be in*
Tested in them to produce an income of £140 ?
Here -j- «= n« of cents, required, for each of which £99| are paid .:
hence the whole sum paid=li? x 99|=£3995.
Ex. 3. If a person were to transfer £29000 stock, fiom the 3| per
cents, at 99, to the 3 per cents, at 90|. what would be the difference in
bis income ?
Here £29000 m the 3| per cents, produces 290 x£3|= £1015 Int.,
and would be sold out for 290 x 99f=£28710 ; this money, invested in
28710
the 3 per cents, at 9 Of, would purchase — '-^- cents., and therefore
28710
would produce, as Int., — -^x 3=£950 85. ; and his income,there-
90^
fore, would be diminished hy £64 I2s,
Ex. €S.
1. The 4 per cents, being at 82|, what must be given for £1000
stock? and what sum would be gained by selling out again at 86^?
2. What income should I get by laying out £1188 in the purchase
of 3 per cent, stock at 81 ?
3. If I lay out £3000 in the 3 per cents, when they are at 84|, what
income should I thence derive?
4. A person having £4200 invests it in the 3^ per cents, at 90 j find
his income.
5. What is the price of stock per cent., when a person can purchase
£2766 13*. 4d, for £2490?
6. What sum must be invested in the 3 per cents, at 94j, to yield an
annual income of £500?
7. How much stock at 92i can be bought for £494, a commission of
I per cent being charged on the stock purchased ?
8. What is the cost of 850 Bank Annuities at 90f, | per cent, being
paid for brokerage? And what sum would be loat Vj ^OXyt^^ ^"viS*
at 89^?
12
1 12 PROFIT AND. LOSS.
9. If I lay out £1030 in the 3| per cent«. at 96, what should I lose
hy selling out at 95?
10. If a person lays out £4650 in the 3| per cents; at 93, what will
be his loss of property by the stocks falling | per cent. ?
11. What would be the difference in income, made by the transfer of
£5000 stock from the 3 per cents, at 72 to the 4 per cents, at 90?
12. A person transfers £11000 from the 4 per cents, at 92 to the 5
per cents, at 110; what is the difference in his income?
13. What would be the difference in annual income from investing
£3450 in the 4 per cents, at 92, and the 3j per cents, at 69?
14. A person invests £18150 in the 3 per cents, at 90|, and, on their
rising to 91, transfers it to the 3| per cents, at 97|: what increase does
he make thereby in his annual income?
15. If I lay out £1 1 10 in the 4 per cents, at 92|, at what price should
they be sold to produce a gain of £100 ?
1 6. In which is it most advantageous to invest, in the 3 per cents, at
89i, or the 3| per cents, at 98 J?
17. A sum of £3750 was sold out of the 3 per cents, at 95, and put
at compound interest for 2 years at 4 per cent. ; the amount being laid
out in the 3| per cents, at 104, find the alteration in income.
18. A person has £1000 in the 3| per cents.; how much must he
have also in the 3 per cents, that his whole income may be £200,
and what sum would he realise by selling out at 83| and 77i re-
spectively?
19. A sum is laid out in the 3 per cents, at 89|, and a half-year*s
dividend received upon it; the stock being then sold at 94|, and the
whole increase of capital being £54, find the original sum laid out.
20. The sum of £1001 was laid out in the 3 per cents, at 89f, and
a whole year's dividend having been received upon it, it was sold out ;
the whole increase of capital being 72 guineas, find at what price it was
sold out.
103. Profit and Loss. — The method of treating ques-
tions of this kind will be best learnt from the following
Examples.
Ex. 1. If tea be bought at 55. 6d. per lb., and sold at 6s, Sd., what is
the gain per cent.?
Here the gain on the prime cost, 5s. 6rf., is Is. 2d. ; hence we have
5s. ed. : £100 :: Is. 2d. : the Ans.
which is found by the usual method to ^le £^\ 4s.
► 10.
PROFIT AND LOSS. 113
Ex. 2. If bar-iron, which cost in making £2 U. Sd, pet cwt., be sold
at a loss of 5| per cent, what price did it fetch per cwt.?
Here bar-iron, which cost £100, would only have sold for £100 £5{
»£94§; hence we have
£100 : £2 U. Sd. : : £94f : the Ans.
which is found by the usual method to be £l 19«. Sjd
Ex. 3. If 5 per cent, be gained by selling 125 yards of cloth for £95,
what was the prime cost per yard?
Here, if the cloth had sold for £105, the prime cost would have been
£100; therefore the selling price per yd. being ■ ^ 2^., we have
123
£105 : ?il£2*. :: £100 : ??-^s. = 14*. 5frf. Ans.
125 21 ^
Ex. 4. If 4 per cent be lost by selling linen at 2«. 9d. a yard, at what
price must it be sold to gain 10 per cent?
Here, cloth which would have cost £100 would have been sold for
£96 at the first price, and for £110 at the second; we have, therefore,
£96 : 2«. 9d. :: £110 : second price = 3«. l^.
Ex. 67.
1. How must nutmegs, which cost 185. Od, per lb., be sold, so as to
gain 16 per cent.?
2. If tea be bought at 28. lid. per lb., and sold at 3^. 7d., what is
the gain per cent. ?
3. A merchant, by selling sugar at £1 I6s. 6d. per cwt., loses 18
pep cent. ; what was his prime cost ?
4. If cheese, which was bought at £3 45. 7d, per cwt., be sold at
£3 125. id., what is the gain per cent.?
5. If iron, raised at an expense of £4 58. 3^J. per ton, be sold at
£4 198. 9d.f what is the gain per cent?
6. If I buy 2048 yards of linen at 35. 2\d. per yard, and sell
the whole for £359 65. Sd. ; required the whole gain and the gain per
cent.
7. If hemp cost £48 75. 6d. per ton, and be sold at £43 per ton,
how much per cent is lost, and how much is lost in the sale of 39 tons,
3 cwt?
8. If 64 ells of lace cost £l 1 5, at what price per yard must it be sold,
so as to gain 18 per cent?
9. A plumber sold 96 cwt. of lead for £109 25. 6</., and gained at
the rate of 12j per cent.; what did it cost him pet tvjV.*^
10. On the sale of 112 yards of silk Ye\\et at\^8, ^d, ^^ i^^cSl, ^
114 PROPORTIONAL PARTS.
merchant loses £10 145. Sd. ; find the prime cost of the whole, and the
loss per cent.
11. If teas at 2^. Qd.j 35. Zd., and 25. id. be mixed in equal quanti-
ties, and the mixture sold at £16 165. per cwt., what will be the gain or
loss per cent. ?
12. A person has ^th of a ship, worth £6600, and insured for 91 J
per cent, of its real value; what damage would he sustain in case of its
being lost.'
13. What was the cost of printing 500 copies of a book, which was
sold for 55., if the expense of sale was 34 per cent., and the author's
profit £37 155. upon the whole?
14. If 5| per cent, be gained by selling butter at £5 55. 6c?. per cwt.,
what will be the gain per cent, by selling it at I5. 3d. per lb.?
15. If 8 per cent be gained by selling 218 yards of cloth for £92 135.,
at what price per yard must it be sold, so as to gain 17 per cent ?
16. A person buys 50 reams of paper, which he thought to sell at
£1 25. 6d. per ream, making 8 per cent profit on the prime cost ; but,
5 reams being damaged, what did he gain or lose per cent, by selling
the remainder at the same rate ?
17. A person buys 4 cwt of goods for £15, intending to gain 12 per
cent by the sale; but, a guinea's worth (at this calculation) being
damaged, at what price should he sell per cwt, to gain as much upon
his whole outlay as he intended ?
18. Bought 236 yards of cambric at 75. 10|«?. per yard, and sold one-
fourth at IO5. 3(/., one-third at 85. 6d., and the remainder at 75. per
yard; what was the gain or loss per cent upon the whole outlay?
19. If eggs be bought at the rate of 5 a penny, how many should be
sold for 7d., to gain 40 per cent. ?
20. A person purchases pins, 18 in a row, and sells them, 1 1 in a
row, at the same price ; how much is his gain per cent on his outlay?
There are various examples depending upon the following Rule, the
method of treating which will be best explained in the instances below
given.
104. Proportional Parts. — To divide a given quantity
into parts which shall have to each other given ratios,
E/ULE. Form fractions whose common den^ is the sum of
the numbers expressing the ratios, and the nnm" the sepa-
rate numbers themselves ; and take these fractions of the
given qnantitjr : they will be the parts required.
PROPORTIONAL PARTS. • 115
Ex. 1. Divide 75 into two parts which shall have the ratio of 2 : 3.
Here the fractions are § and f, and the parts reqoired are f of 75 ==30,
and f of 75 =45, which are plainly in the given ratio.
The reason of the Hale is evident, since the sum of the nam" makeb
up the den', and therefore the sam of the fractions makes up uniti/y i e.
the sum of the parts makes up the loliole of the number; while the parts
themselves, having a common den**, are in the ratio of their nam".
Ex. 2. Gunpowder is composed of 76 parts of nitre, 14 of charcoal,
and 10 of sulphur: how much of each of these will be required for a cwt
of powder?
Here the fractions are ^=|f, ^=A, ^=A» and the parts are
8q. l^lbs., ,15||lbs., and ll^lbs. respectively.
Ex. 3. Divide £1000 among A, By C, so that A may have half as
much again as B, and B a third as much again as C
Here, representing (7s part by 1, B*s is 1|, and ^'s 1J + | of 1^=2;
and, therefore, the parts are to be as the numbers 2, li 1, or 6, 4, 3.
Hence the fractions will be ^, ^, ^; and the parts required
£461 10«. 9^., £307 13s. 10^., £230 I5s. 4^
N.Bi — It will be found most convenient, where there are many frac-
tions with the same den', to find the part corresponding to that den'
with num' unity, and then multiply this successively by the num" of the
different fractions; thus we should find ^ of £1000, and then multiply
this by 6, 4, 3, respectively.
Ex. 4. Af B, and C form a joint capital for conducting a business, of
which A contributes £500, B £650, and C £700. At the end of a year
the profits are £555; what share should each receive?
Their shares should evidently be in the ratio of their contributions of
capital, i.e. in the ratio of 500, 650, 700, or of 10, 13, 14; hence the
fractions are §?, U, |f, and since 3V of £555 =£15, we have the shares
required £150, £195, £210.
Ex. 5. A begins business with a capital of £800, and, at the end of
3 months, takes B into partnership, with a capital of £1000; at the end
of another 6 months they divide their profits, £330 ; what should each
Teceive?
Here A contributes £800 for 9 months, and B £1000 for 6 months >
and the interest of £800 for 9 months = interest of £800 x 9 for 1 mo.,
and the interest of £1000 for 6 months = interest of £1000x6 for
1 month ; hence the value of -4*s and B^s outlay may be represented by
the products 800 x 9 and 1000 x 6, or 7200 and 6000 respectively, and
their shares of the profits must be in this ratio = that of 6 : 5 ; l\^\v^<& ^%
shares^ of £330»£180, and ^s share=^ oi £^^Q»£\^Q.
116 PROPORTIONAL PARTS.
K.B. — ^It appears, as in the above Ex., that the values of sams
employed in businesSj &c., for different times are proportional to the pro-
ducts of the sums by the times, or rather of their numerical values, the
sums being expressed in the same den^ aud so also the times.
Ex. 6. A and B enter into partnership, A contributing £500 and Zi
£300 ; ^t the end of 9 months they take in C as partner, who brings into
the concern a capital of £1000. The profits, £2000, being divided al
the end of another 9 months, what shares did they each receive?
Here, as in Ex. 5, at the end of 18 months, the shares of capital
supplied by A, B, C, respectively, may be measured by the numbers
500 X 18. 300 X 18, 1000 x 9, or 5, 3, 5 respectively: hence the fractions
will be A, ft, A; and since ^ of £2000==£l53 16*. ll^c/., their
shares of profit will be £769 4*. 7^., £461 10*. 9^., £769 4*. 7^.,
respectively.
Ex. 68.
1. Divide 1065 into parts, which shall be to each other in the ratio of
3, 5, 7 ; and also into parts which shall be in the ratio of |, |, ^.
2. A, By and C engage in trade, investing capital to the amount of
£128, £176, £192 respectively: their profits amount to £279; what
were their shares of it?
3. How much copper and tin will be required to cast a cannon
weighing 16 cwt. 3 qrs. 11 lbs., gun-metal being composed of 100 parts
of copper and 11 of tin?
4. Divide £153 among five persons in the proportion of the fractions
11111
3» 4' '€* tj» ?•
5. Divide 1400 into parts, which shall have the same ratio to one
another as the cubes of the first four natural numbers.
6. Pure water is composed of 2 gases, oxygen and hydrogen, in
tl)c proportion of 88.9 to 11.1; what weight of each is there in a cubic
foot ClOOOoz.) of water?
7. Divide £300 among three persons, so that the first shall have
twice as much as the second, and the third twice as much as the other
two together.
8. A works regularly 9 hours a day ; B remains idle the first two
days of the week, and works 6*, 8|, lOf, 12 hours, respectively, on the
other four; what sum should each receive out of £11 12*. 6|(/. at the
month's end?
9. The standard silver coin of this realm is made of 37 parts of
pure silver and 3 of copper, and a lb. Troy of this metal yields 66 shil-
lings; what weight of pure silver is there in 20*.?
10. In England, gunpowder is made of 75 parts of nitre, 10 of sul-
pliur, and J J$ of charcoal; in France, of 77 of nitre, 9 of sulphur, and 14
PROPORTIONAL PARTS. 117
of charcoal: if half a ton of each be mixed, what weight of nitre, sol-
phar, and charcoal, will there be in the compound?
1 1. The standard gold coin of this realm is made of gold, 22 carats
fine, and a lb. Troy of this metal yields 46 J§ sovereigns; what weight of
pare gold is there in 100 sovereigns?
12. If 4 oz. of gold, 17 carats fine [*e« Appendix'}, are mixed with '
3 oz., 1 3 carats fine, how much fine gold will there be in a gold orna-
ment made of the compound, and weighing 3| oz.? (
13. A and B engage in trade, their capitals being in the ratio of
4:5; and, at the end of three months, they withdrew respectively §
and I of their capitals: how should they divide their whole gain, £335,
at the end of the year ?
14. A, Bj C join their capitals, which are in the proportion of |, |,
and \ ; at the end of 4 months A withdraws \ of his capital, and at the
end of 9 months more they divide their profits, £284; what should each
receive ?
15. A and B rent a pasture for £75; A puts in 80 sheep and B
100, but at the end of 6 months they each dispose of half their stock,
and allow C to put in 50 sheep to feed; what should A, B, C, severally
pay towards the rent at the year's end?
16. Four parcels of gold, weighing respectively 10, 4, 2, and 4 oz.,
and of 13, 12, 11, and 10 carats fineness, being mixed, what was the
fineness of the compound ?
17. If the preceding be reduced by refining to 16 oz., what will l)e
the fineness of the mass? or if its fineness, when reduced, be 16 carats,
what will be the reduced weight?
18. If 8 oz. of gold, 10 carats fine, and 2 oz., 11 carats fine, be
mixed wirh 6 oz. of unknown fineness, and that of the mixture bo 12
carats, what was the unknown fineness?
19. Af B, C, are sent to empty a cistern, by means of two pumps of
the same bore. A and B go to work first, making 37 and 40 strokes re-
spectively a minute ; but, after 5 minutes, they make each 5 strokes less
a minute, and, after 10 minutes more, A gives way to C, who works at
the rate of 30 strokes a minute. The cistern is emptied in 22 minutes
altogether, and the men arc paid 12^. 7d, for their labour. What should
each receive ?
20. A and B are partners, having each embarked £500 in their
business. At the end of 3 months they gained £300, when A withdraws
£200, and B at the same time advances £200. At the end of the next
3 months, they gained £780, when A again withdraws £200, and B at
the same time advances £200. At the end of the year iVve^j ^e^^st^vs.^
dividing their property, which by losses duniig l\ift \a&\. ^ xaaw'CcA -v^^
x^uced to £400, Wh&t should A and B eacla x^mv^^
118 CHAIN BULB.
105. Chain Rule. — ^When a comparison of several suc-
cessive quantities is made by stating how many of the
second are equivalent to a given number of the first, how
many of the third are equivalent to a given number of the
second, and so forth, and it is required to find how many of
the last are equivalent to a given number of the first, the
answer is conveniently found by the Chain Rule, The fol-
lowing is an example :
What is the value of 20 lbs. of bacon, if 15 lbs. of bacon be equal in
value to 14 lbs. of cheese, and 35 lbs. of cheese equal to 46 lbs. of pork,
if pork be worth 65. Zd. per stone of 8 lbs. ?
In applying the Chain Eule to this question, we first set down the
direct demand — How many pence = 20 lbs. of bacon ? — which may be written
briefly thus : ? pence = 20 lbs. bacon ; then we set down a given quantity
of bacon as equivalent to a given quantity of something else : thus, 15 lbs.
bacon = 14 lbs. clieese; then another given quantity of cheese as equiva-
lent to something else: thus, 35 lbs. cheese =^6 lbs. park; then another
given quantity of pork as equivalent to something else: thus, 8 lbs.
pork = 76 pence. These equations should be placed in successive lines,
as follows :
? pence = 20 lbs. bacon,
if 15 lbs. bacon = 14 lbs. cheese,
35 lbs. cheese = 46 lbs. pork,
8 lbs. pork = 75 pence ;
where it may be observed that the first and last quantities in the state-
ment are of like denomination, viz. pence^ and that the second side of an
equation is always of the same kind and denomination as the first side of
the next equation. The answer for the term of demand (J pence) will
now be found by dividing the continued product of the right-hand num-
bers by that of the left-hand numbers. Thus :
6 ^ \\
!^^xXW6>LW=.5^46 = 230t?. = 19..2rf. An^.
X ^
The reason of the equating and calculating processes will be evident
if we employ unity to express the antecedent of each condition ; thus :
? pence = 20 lbs. bacon,
if 1 lb. bacon = i| lb. cheese,
1 lb. cheese =11 lb. pork,
1 lb. pork = ^ pence ;
OHADf BULB. Il9
for now it is obvious that 20 lbs. bacon = ^x20 lbs. clieese = |fxi|
X 20 lbs. pork= "^5 x Jf ^ it ^ 20 pence.
The most important application of the Chain Bule belongs to what is
called Arbitration of Exchange. — See Note XI.
Ex. 69.
1. If 10 first-class labourers do as much work per hour as 12 second*
class, 14 second-class as much as 16 third-class, 18 third-class as much
as 21 fonrth-class, what number of the first class corresponds to 8 of
the fourth?
2. When 94^ Dutch florins is the exchange for 100 Austrian florins,
and 16 sovereigns are given for 193 J Dutch florins, how many Austrian
florins should be given for 28 sovereigns ?
3. How many lbs. of tea are equivalent to lOg lbs. of butter, when 5
lbs. of tea are equivalent to 14 of coffee, 9 of coffee to 20 of sugar, 10 of
sugar to 6 of cheese, and 10 of cheese to 9 of butter?
4. If 8 sacks of flour be equal in value to 13 loads of straw, 3 sacks
of flonr to 10 sacks of potatoes, 27 sacks of potatoes to 26 cwt. of rice,
and 18 bushels of oats to 5 cwt. of rice, how many loads of straw are
worth as much as 10 bushels of oats ?
5. If 16 pears be equal in price to 25 apples, and 18 oranges equal
to 12 pears, and 20 lemons equal to 27 oranges, and lemons cost 13^. a
dozen, what is the cost of 15 apples ?
6. How many yards of velvet are equal in value to 60 of muslin,
when 25 of muslin are equal to 16 of calico, 21 of calico to 13 of flannel,
40 of flannel to 27 of linen, 68^ of linen to 28 of silk, and 47 of silk to
36 of velvet?
7. How many pounds sterling will be the value of 1000 rupees, when
15 rupees are worth 7 American dollars, 5 dollars worth 26 francs, and
101 francs worth £4 ?
8. If 4 quarters of oats be worth 3 quarters of barley, 14 quarters of
barley worth 11 quarters of wheat, 27 quarters of wheat worth 32 bags
of rice, 24 bags of rice worth 67 sacks of potatoes, and 2 sacks of pota-
toes weigh 3 cwt., what quantity of potatoes is equivalent to 63 hxisheU
of oats ?
9. When J of a lb. of tea is equal in value to ^ of a stone of mutton,
and f of a stone of mutton equal to 3 lbs. of coffee, and J of a lb. of
coffee equal to ^ of a lb. of beef, how many lbs. of beef are equivalent to
^Olbs. of tea?
10. If an ounce troy of standard silver, of which 37 in 40 parts of
the whole are fine, be worth bs. 1J<?., and copper worth 5 guineas per
cwt., what is the ratio of the value of fine silver to thai oi coys^^*^
120 SQUARE ROOT.
106. Square Root. — The square, or 2n(l power, of a
given abstract no. is the product of the given no. by itself.
Thus the square of 7, or, as commonly indicated, 7^
=7 X 7=49 ; and 7 is called the square root of 49.
The sign of the square root is v/, a corrupted form, of the
initial letter of the Latin word rad^x^ root ; thus we write
^40= 7.
Few numbers, comparatively, are 'perfexit squares ; as
may be seen by the intervals of the numbers 1, 4, 9, 16, 25,
36, 40, 64, 81, which are the squares, respectively, of 1, 2,
3, 4, 5, 6, 7, 8, 9, and which indicate that every perfect
square must have 1, 4, 5, 6, or 9, as its last significant
figure.
107. Now, as the square root of 49 is 7, because 7 x 7=49,
so the square root of 186624 is 432, because 432x432
=186624; but, while simply from recollection of the ordi-
nary Multiplication Table it is easy to tell what is the
square root of 49, a process somewhat complex is requisite
to extract from 186624 the square root of that high nunaber.
We proceed to exemplify tlie method of extracting the
square root of a large number, referring for 'proof of the
method to the chapter on Involution and Evolution in
Colcnso's Algebra.
Ex. 1. Extract the square roots of 186624, 77841, 9659664.
166624(432
16
:?764U279
4
5669664(3108
9
83)266
249
47)378
329
61) 65
61
862) 1724
1724
549) "4941
4941
6208) 49664
49664
Here wo first place a dot over the last figure, and then over every
second figure, reckoning from it ; by which means the number will be
divided into periods, as they are called, consisting each of two figures,
except the first, which (when the number of figures in the giyen number
is odd) will evidently consist of only one figure.
We then take the nearest square n° not greater than the first period:
this is 16 in the first of the above instances* and we set its square root, 4i
SQUARE BOOT. 121
as the first figure in the root ; we then subtract its square, 16, and bring
down the next period, 66.
We now set the double of the first figure in the root, 8, in a loop, as
divisor, to the left of the rem', regarding it, however, as standing for 80,
not for 8, since we shall presently have to set another figure after it.
Dividing the rem' by this div, 80, we set the quotient, 3, as the second
figure both in the root and also in the div' : then, multiplying the 83 by
3, we subtract the product, and take down the remaining period, 24.
To form, the next div', we double the last figure of the preceding
one, making 86, which (as before) we regard as 860, and proceed
exactly in the same manner ; and if finally, as here, we find there is
no rem', we may conclude that we have found the exact square root.
In the 2nd instance, notice (i) that the second rem', 49, is greater
than the div% 47 ; this may sometimes happen, but no difficulty can
arise from it, as it would be found that, if instead of 7 we took 8 for
the second figure, the subtrahend would be 384, which is too large :
And (ii), that the last figure, 7, of the first div% being doubled in ordet
to make the second div, and thus becoming 14, causes 1 to be added to
the preceding figure, 4, which now becomes 5.
In the 3rd instance, we have an intermediate cypher in the root.
Ex. 2. Extract the square roots of 1000, 2, 1.6, .002.
A/l0O0-O0 = 3r6&c. ^/5 = r41&c. ^/l•60 = l•26&c. v/-0020 = -0447&c.
9 11 16
61)100
61
24) 100
96
22) 60
44
84) 400
336
626)3900
3756
281) 400
281
246) 1600
1476
887) 6400
6209
144 119 124 191
In the 1st instance, we find there is a rem', 29, when we have made
use of the last period of the given number, 1000 ; but we may continue
the operation as long as we please in such a case, by setting a decimal
point after the given number, and annexing cyphers as decimal places ;
and for every period of two cyphers thus formed we shall obtain a
decimal figure in the root.
The same is true of the 2nd instance, except that we have not taken
the trouble to set down the extra cyphers at the end of the given number,
though we have taken them down as required, and set the decimal point
in the root.
In the 3rd instance, it is to be noticed that the first dot must always be
placed on the last figure of the integral part of any number, i.e. on the
one nextbeforfe the decimal point, and then on every second ^^\«v5,\scl
each side of it Of course, in the 4th instance, l\ifc ^^va^ TL«»x\«Xsst^
122 SQUAEE BOOT.
the decimal point, though not expressed, is 0. And, in both these, we
have had to annex one cypher to the original namber, to complete its
points.
In all such cases the square root can never be exactly obtained ; but
by annexing cyphers, it may be ascertained to as many places of deci-
mals as we please. Such roots are called irrational or surds,
1 fiQ ^T T *»
Ex. 3. Extract the square roots of , — , —, and •-.
^ 289 64 12 7
(\\ 169^ 13x13 . /
^•^ 289 17xl7"V
169 -v/169 13
289 V289 17
n\ \ /37_ a/37_ a/37 _ 6-082762 + _.y
^'''^ V^"^^"'8— 8 -760345 + .
Or, . /^=a/-578125 = -760345 + .
V 64
^ ' V 12 V 36 6 6 « «' -r.
Or, a/^«a/-583333.. = -76376 + .
Or, ^1 = A/-71428571 + = -845154.
108. In any right-angled triangle, as abc,
li p denote the length of the perpendicular,
b that of the base, h that of the hypote-
nuse or side opposite the right angle ; then
^2_j_p2--.^2^ Qj, h'^ — b'^:=p\ or h^—p^=b^,
(Euclid, I. 47.)
Ex. 1. Given 6 = 20, ;? = 21, to find h.
>v/(202 + 212)= y(400 + 441)= a/841 = 29. Ans.
Ex. 2. G-iven A = 29, 6 = 20, to find p.
a/(292-202)= a/(841-400)= a/441 = 21. Ans,
Note. — It is useful to remember that the difference of the squares of
two nos. may be found by multiplying the sum of the nos. by their
difference; thus 292-202 = 49 x 9 = 441.
Ex. 3. When p is 21 inches, and ^ is 9 inches longer than 5, what
is the length of 6 ?
Here the sum of h and b x their difference 9 —p^ = 441 ; and hence
441 -r 9 = 49, the sum of h and b ; therefore b = ^(49 - 9} = 20. Ana,
Ex. 70.
Extract the square roots of —
/. 6329 and 8836. 2. 34225 and ISl^U. ^. &^UU and 350464.
CUBE ROOT. 123
4. 95481 and 249001.
5. 348100 and 6512490000.
6. 37491129 and 16949689.
7. 3534400 and 65561409.
8. 99960004 and 24088464.
9. 119550669121 and 368451428004.
10. 8, 20, and 363.
11. 35120 and 8837.
12. 134909*29 and 650506*7716.
13. 6663-114 and 27-773.
14. -225 and 51-12965025.
15. '012012 and -00158404.
16. -000082355625 and '021.
17 B29 18 anA 139
18. ^.^Vai»d3^.
19. 287i 6136i, and 367f.
20. A, :^*,and3-. *
303' -155 * 6-f
21. l§of(4| + 5|),andl+|-| + l-i.
22. How many links long is a square field containing 8 ac. 2 ro. 9 po.?
23. Find the side, and also the diagonal, of a square having the
same area as a rectangle 43 ft. 5 in. long and 34 ft. 7 in. broad.
24. M and N start together at B, to walk to another point C,
1332 yards north from B ; M takes the direct road BCy N goes first to
At a point west from By and then straight to C, his whole journey being
2738 yards. Howfar is it from ^ to ^ ?
109. Cube Root. — The cube root of a given number is
that number wbicb when multiplied by its square produces
the given number. Thus, using %/ as the sign of the cub6
root, we have %/ 512=8, because 8 x 8 x 8=512.
The first nine numbers are the respective cube roots of
"1, 8, 27, 64, 125, 216, 343, 512, and 729.
The method of eictracting the cube root of a large number
is much more complex than that required for the square
root, as will appear from the following example. A proof
of the method will be found in the chapter on Involution
and Evolution in Golenso's Algebra,
124
CUBE ROOT.
Ex. Extract the cube root of 80677568161*
^80677566161=4321
64
123
1202
12961
4800
369
6169
16677
16507
654700
2584
657284
1170568
1114568
65987200
12961
66000161
66000161
66000161
^
Here -^e first
divide the number
into periods by
placing a dot over
the last figure, and
then over every
third figure begin-
ning from it. Then
we take the nearest
cube n® not greater
than the first peri od,
80 ; this is 64, and
we set its cube root, 4, as the first figure in the root ; then, subtracting
its cube, 64, we bring down the next period, 677. We now set the
triple of the first figure of the root, 12, at some distance to the left of
the rem'; (there is 123 in the sum, but the 3 will be accounted for by
and by ;) then we multiply this triple by the first figure of the root, and
place the product, 48, between 12 and the rem*", annexing two cyphers
to it.
"We now divide the rem' by this 4800, and set the quotient, 3, as the
second figure in the root, and also after the 12, making 123 : now we
multiply this 123 by 3, the second figure in the root, set the product, 369,
under 4800, add them up, multiply the sum, 6169, by the second figure
in the root and subtract the product, 15507. We bring down the next
period, 668, and have now to fprm the two quantities to the left of it
The first is obtained by tripling the last figure, 3, of 123, which gives 129
(the final 2 in 1292 will be accounted for when the next figure in the
root is found) ; and the other quantity, 5547, is found by adding 9, the
square of the second figure in the root, to the two preceding middle
369
lines e-aQ* We now add two cyphers, and repeat the whole process
described in this paragraph.
The remarks made above with respect to surd square-roots apply
also to cube-roots : thus, '01, 24*1 would be pointed for the cube-root
•010, 24-100.
Ex.71.
Find the cube roots of —
1. 185193 and 405224.
2. 21952 and 6859000.
3. 4330747 and 35287552.
4. 94818816 and 961604803000.
CUBE JlOOT. 125
6. 529475129 and 111423515328.
6. 261775532773 and 176369715712.
7. 357759791-299 and 050243409.
8. 50000 and 527*71.
9. 8047 and 5678'9.
10. fandSOj.
11. A box is 3 ft 5 in. long, 1 ft. 8 in. wide, and 14 inches deep.
Hequired the edge of a cubical box of the same capacity.
12. The volumes of spheres are to one another as Uie cubes of their
diameters. If, therefore, the Sun be 1^ million times as large as the
Earth, and the Earth's diameter be 7912 milest how many miles will the
Sun's diameter measure?
126
MISCELLANEOUS EXAMPLES. 7a.
1. The circumference of a coach- wheel being 16j ft., how often will
it turn round between London and Oxford, a distance of 59 miles ?
2. If a person's estate produce £400 a year, and the land-tax be
assessed at 2$, 9d, in the pound, what is his net annual income?
3. Eeduce -^^ to its lowest terms, and £1 Ids, Gd. to the fraction
of a guinea ; find the value of £^ of half-a-guinea, and add together |,
fofi?, lf,and3H-2|.
4. Divide 21| guineas equally among 12 men.
6. What is the rent of 145a. 1r. 32p. of land, at £10 55. dd, per
dcre ?
6. The produce of a farm one year was 150 quarters, which were sold
at 585. a qu. ; in the next year the price of wheat fell to 485., but the crop,
being plentiful, produced on the sale the same amount as before : of how
many quarters did the second crop consist ?
7. A straight plank is 3^ in. thick, and 6J in. broad ; what length
must be cut off so as to contain 6 J cubic feet of timber?
8. A person holding 50 shares in the London and North-Western
Railway, sells out at 170 ; what income would he have by buying into
the 3i per cents, at 93J ?
9. If 5 lbs. of tea be worth 12 lbs. of coffee, and 7 lbs. of coffee worth
20 lbs. of sugar, and 14 lbs. of sugar worth 75. l|e^., what is the worth
of 9 lbs. of tea?
10. A common pasture containing 54a. Se. SS^p., another containing
39a. 13Jp., and a third containing 54iA., are to be divided into 60 equal
parts, after deducting from the whole 1 1a. 2|r. for tithes ; of how much
does one part consist ?
11. Eind the square root of 370881, and the side of a square con-
taining 7367 sq. ft. 52 in.
12. If the produce of wheat be tenfold of the seed, how many
quarters can be obtained from one grain in 10 years, supposing there to
be 7580 grains in a pint ?
13. If I lose l^d. in 35. 4d., how much do I lose per cent. ?
14. In the centigrade thermometer the freezing point is zero, and the
boiling point is 100° ; in Fahrenheit's the freezing point is 32°, and the
boiling point is 212° ; what degree C. corresponds to 68 F. ?
15. How much water must be added to a cask, containing 40 galloDB
ofspinta at 135. %d,, to reduce the price to 105, ^, ?
HiSCELLANtlOTTS fiXAMPLilS. 127
16. A bill for £100 lias six months to run, and the holder has it dis-
counted at 5 per cent., and receives £97 10^. ; how much less than his
due does he receive ?
17. Find the value of f of a guinea ; reduce 25. S^d. to the fraction
of a pound, and 1 hr. 71 min. to the fraction of 1 da. 6 hrs.
18. A person invested £1000 in the 3 per cents, at 90| ; but the price
rising to 91 J, he sold out, and invested the proceeds in the 3| per cents,
at 97^ : find the increase in his income.
19. Find the square root, and also the cube root, of 9595 IffJ.
20. A general levies a contribution of £870 on four villages, con-
taining 250, 300, 400, and 500 inhabitants respectively ; -what must they
each pay?
21. A can do a piece of work in 10 days, which B could do in 13 ; in
what time would they do it together ?
22. A stationer sold quills at ll5. a thousand, by which he cleared |
of the money ; what would ho clear per cent, by selling them at 13^. 6d,
a thousand ?
23. Keduce 0^, 17^ + A + H^Jl, 2I|-||, | of ? x ^ of i| of g,
and 6347 -5- 2J, to their simplest forms.
24. Divide the value of 79 florins between A and B, giving A half-
a-erown more than B.
26. Three persons rent a piece of land for £60 lOs.; A puts in 5 sheep
for 4J months, B^ 8 sheep for 5 months, and C7, 9 sheep for 6| months :
what must each pay of the rent ?
26. What is the present worth of £75, due 15 months hence, at 5 per
cent.-?
27. If A can do a piece of work in 10 days, and A and B can do it
together in 7 days, in what time would B alone do it ?
28. Find the cube root of 133354510.
29. Divide £16 0^. lOe^. among 4 persons in the proportion of the
fractions |, |, J, i.
30. Divide 1037 into two parts, which shall have to one another the
ratio of the sum of 7*625 and 5*375 to their difference.
31. A cistern has two pipes, by one of which it may be filled in 40
min., and by the other in 50 min. ; it has also a discharging pipe, by
which it may be emptied in 25 min. If all these three were open toge-
ther, in what time would the cistern be filled ?
32. There is a number which, when divided by | of j of 1|, "will pro-
duce 1 ; find its square.
33. If a person lend me 1296 guineas for 125 days, how long should
I lend him £1620 to requite the fiivour?
34. Find the square roots of 9*21677 and 921677.
85. If 6 men will dig a trench, 15 yds. long ao^ 4 \A^»i^SxL^ ^sf^
k2
128 MISCELLANEOUS EXAMPLES.
of 9 hours each, in hoTV many days of 8 hours each will 8 men dig a
trench 20 yds. long and 7 broad ?
36. Eeduce 1 3^. 7^d, to the decimal of a pound, and f of Is. 5^. to the
fraction of half-a-crown ; divide 1001 by 390626, '1001 by '000390626,
and 10*01 by 390'625.
37. The cost price of a book is Ss, 9d. ; if the expense of sale be 6
per cent, upon this, and the profit 24 per cent., what would be the retail
price?
38. If the Sun moves through 360® in 366 days 6 hrs. 48 m., how
many minutes and seconds will he pass through in a day ?
39.' Divide £16 among 10 men, 13 women, and 26 children; each
man to receive twice as much as each woman, and each child half as
much as each woman.
40. There is a fraction which, when multiplied by the cube of 1|, and
divided by the square root of 1|, produces f ; find it.
41. A floor, 24 ft. 4 in. broad and 96 ft. 6 in. long, is to be laid at
l^., per square foot ; find the cost.
42. A sells to i? I of i of I of 30 sheep for ^ of ^ of f of £210;
what was the average price of each sheep ?
43. The estate of a bankrupt, £21000, is to be divided among four
creditors, whose debts are, ^'s to ^s as 2 : 3, ^s to C*s as 4 : 5, Cs to
D'a as 6 : 7 ; what must each receive?
44. A cubic foot of water weighs 63 lbs. ; what is the weight of water
in a vessel 1 ft. deep, 16 ft. 7 in. long, and 8 fb. 4 in. wide?
46. The profits of a mine for one year amounted to £3296 13«. 5^.,
and a person holding 14 shares received for his dividend the sum of
£1026 128. 7ld. ; how many shares were there in all?
46. If the price of gold be 4 guineas an oz., what is the cost of a gold
ornament weighing 3 oz., of which 18 parts out of 24 are pure gold;
allowing Zs. 4d. per oz. for the value of alloy, and 26 per cent, upon the
whole for expense of workmanship ?
47. Find the square roots of '064 and 26-123466790.
48. What is the price of a piece of timber, of which the length,
breadth, and thickness are respectively 23 fb. 9 in., 2 ft. 4 in., and 2 ft,
at Q\d, per solid foot?
49. If 90 degrees correspond to 100 French grades, how many
degrees and how many grades are there in the sum of 36*46 degrees^
and 36*46 grades?
60. A man can reap 302^ square yards in one hour ; in what time
"will 3 such men reap 2| acres ?
61. A fiirmer gave for a horse a bill of £166, due 8 months hence^ at
4§ per cent., and sold him at once for £180 ; required his gain per cent-
62. ^ can do a piece of work in 3 days, B can do thrice as much in
UISCELUU^EOUS EXAMPLES. 129^
8 days, and C fire times as much in 12 days : in what time would they
doit together?
53. If a tradesman marks his goods 20 per cent, above the cash price^
what ready money would he take for an article marked 26s. ?
54. If 6 men can earn £20 in 21 days, when the days are 12 hrs.
long, how much can 4 men earn in 35 days, when the days are 10 hrs.
long?
55. If 45 bricks will pave a square yard, how many will be wanted
for a space 34 fb. long and 14 ft. wide, allowing for a path, 2 feet wide^
all round?
56. Keduce 3|«. to the decimal of ^\ of a guinea ; and find the values
of -232 of a cwt., and 4-01? 1 of a mile.
57< A gentleman had 5 sons, to whom he left £3750 in cash, and
two bills of £151 each, due at the end of two and three months respec-
tively ; the eldest son had by the will J of the property, and, taking
charge of the whole, he paid the others their shares, which were equal, in
cash. What would these be, reckoning interest at 4 per cent. ?
58. Find the sq. root of 39*0625, and the cube root of 2116-874304.
59. What is the annual interest on £76978, bought into the Danish
8} per cents, at 77 ? and what sum would be gained by selling out
at77i?
60. It is desired to cut off an acre of land from a field 1 5^?. in
breadth ; what length must be taken ?
61. Express a degree (69^ miles) in metres, when 32 metres are
equal to 35 yds.
62. At 9|^. per sq. yd. what is the cost of painting a room which is
24 yds. round, and 10 ft. 4 in. in height ?
63. Find the difference between V§ and ^|.
64. What is the alteration in income made by transferring £10000
from the 3 per cents, at 92 to the 4 per cents, at 110 ?
65. Divide 4J into two parts, one of them to be 4^ times the other.
66. A plate of gold, 3 in. square and \ in. thick, is extended by
hammering so as to cover a surface of 7 sq. yds. ; find its present
thickness.
67. I bought 171 gallons of brandy in bond for £79 Ss. 4d., and on
taking it out paid duty equal to 112| per cent, of the bonded value;
what was the duty per gallon ?
68. Compare the interest on £350 at 4| per cent., with the interest
on £450 at 3| per cent., for one year.
69. The top of a ladder, 2G J feet long, exactly reaches the top of a
wall, when its foot rests at a point 7J feet from the bottom of the wall.
Determine the height of the wall, and how far from the bcAXoisi^Qiv^
ladder would rest, if it were 28 fe§t long.
130 MISCELLANEOUS EXAMPLES.
70. A drawing-room, 36 ft. 10 in. long and 23 ft. 2 in. wide, is sup-
rounded with a cornice 3i in. wide, the gilding of which cost £4 1 Is, 10|rf. ;
how much was that per square foot ?
71. A steward receives for his landlord £1987 of rent, and disburses
one-fifbh ; he pays his landlord £105 12^., and the remainder is inyested
in an estate at 30 years* purchase : find the rent of the estate.
72. Reduce ^ of half-a-crown to the fraction of half-a-guinea,
and 6^. Sid. to the decimal of a £ ; find also the yalue of f of | of
£6666 Us. 4d.
73. "What is the yearly interest on £1 127 bought into the 4 per cents,
at 92?
74. Find the value of £1368 7s. 6d. sterling in dollars and cents, a
dollar being equal to 100 cents, and to 4$. 4d. English money.
75. A sum of £333 3^. S^. is to be divided among 4 persons, whose
shares are to be in proportion as 1, 2, 3, 4 ; find the share of each.
76. The circumference of the Earth in the lat. of London is 15120
miles ; find the distance between two successive meridians of longitude^
and the space passed over by the Sun in his apparent daily motion in a
minute.
77. If a person accepts £247 Is. Sd. as present payment of £252 Os. 6d,
due four months hence, at what rate per cent, does he allow discount?
78. Divide 13s. l^. into six parts, each succeeding part to be 6|<^.
more than each preceding.
79. How much stock, at 93^ per cent., can be purchased for £540, a
commission of Jth per cent, being charged on the stock purchased?
80. If either 5 oxen or 7 horses will eat up the grass of a close in 87
days, in what time will 2 oxen and 3 horses eat up the same ?
81. The sum of £3 13s. 6d. is to be divided among 21 men, 21 women,
and 21 children, so that a woman may have as much as two children, and
a man as much as a woman and a child ; what will each man, woman,
and child receive ?
82. A sells to 5 I of f of ^ of a package of tea, which weighs ? of J
of 1 cwt. 21 lbs. at 3s. 6d. per lb. ; what did it come to?
83. How many revolutions will a carriage- wheel, whose diameter is
a yard, make in a mile, the ratio of the diameter to the circumference
being 1 : 3*14159?
84. A cistern can be filled by two pipes, A and B, in 4 min. and
5 min. respectively, and emptied by C in 2| min. A is opened for
2 min., and then A and B together for 1 min. more, when C is also
opened. In what time would tiie cistern, which now contains 861 gals.,
be full? and how many gallons would have passed through A and B
respectively?
8o. What J8 the yearly interest ou £27225, bought into the 8} per
cents, at 07^ ?
MISCELLANEOUS EXAMPLES. 131
86. Express inits simplest form i|-l| + i|-ii; and add tx)gether |
of a guinea, ^ of a crown, and ^ of 7«. 6«?., and reduce the result to the
decimal of 16«.
87. Find the simple interest on £325 I6s. Sd., for 5 months, at 4|
per cent.
88. If 18 men eat I6s, worth of bread in 3 days, when wheat is at
bis., what yalue of bread will 45 men eat in 27 days, when wheat is
at 45«. ?
89. What length of paper, 22| in. broad, will be used for a room
2.1 ft. 9J in. long, 15 ft. 7 in. broad, and 8 ft. l|in. in height? and what
will it cost at Is, Sd. a yard ?
00. Find the value of 36*42 tons of coal, at 17*. 7^. per ton ; and
the difierence between f x ft x il of l^*-* and ^ of f of £3 1 Is, Qd,
91. The 3 per cents, are at 85|; what price should the 3| per
cents, bear, that an investment may be made with equal advantage
into either stock ? And what income would be derived by so investing
£5000?
92. A farm lets for £92 per annum : the tenant pays for 2 yean?
occupation, with interest accumulating at 6 per cent. ; the landlord
pays J the amount for repairs of house, J of this for repairs of barn, and
£2 38, 4d, for other expenses : find the balance.
93. What will be the cost of painting a room at 9^. per square yard,
if the sides are each 19 ft. 10^ in., the ends 16 ft. 1| in., and the height
10 ft 3 in.?
94. Express 1618| Eng. miles in degrees (a degree = 69^ miles):
find the values of f of £2 7^. 8|^., and of ft of £l 6s. Sd., and reduce
their difference to the decimal of £20.
95. Twenty-six wedges of gold, weighing in all 33 lb. 3 oz. 7 dwt.
4 gr., are to be coined into sovereigns : find the weight of each wedge,
and the number of sovereigns coined from the whole, at the rate of Z^
sovereigns per oz.
96. How many feet in 150 must a road 10798 feet long rise, to be
carried from a plain to a hill 463 feet in perpendicular height?
97. A gentleman selling a mortgage of £4410, for which he received
5 per cent, interest, bought into the 3| per cents. Bank Stock at 70 ;
after receiving the interest for 5 years, on the stocks rising to 75, he sold
out. What was his gain upon the whole transaction, over what he would
have received had he continued the mortgage ? "
98. What is the present worth of £325 I6s. Sd., due at the end of 5
months, at 4| per cent. ?
99. Find the square roots of 6242i and 1438*237, and the cube roots
of '000328509 and 27054*036008.
100. If 40 men in 7f days can dig 3 rectanguXaj ^q\^ ^^\^^ ^3^a.
132 meCELLAKEOUS EXAMPLES.
by 130; how long will 37 men be digging 5 fields, e&ch 120| yd&,
by 90?
101. If 3 men, 5 women, or 8 children, could do a quantity of work
in 26| hours, in what time will 2 men, 3 women, and 4 children com-
plete it?
102. A person, leaving Paddington at 13 minutes before 2, P.1L,
trayels the first 162 miles at 27 miles an hour, the next 121 miles at 9}
miles an hour, and the last 27 miles at 8 miles an hour : when will he
reach his destination, Penzance ?
103. How many square yards are therein a parade, 864 fb. 3 in. long
and 62 ft. 6 in. broad ?
104. A met two beggars, B and C, and, having in his pocket
(3^+4|) of (lOf -r 7|) of ^ of a moidore (27«.)» g»^e B J of | of that
sum, and C f of the remainder ; what did each receive ?
105. What is the present worth of £1147 10s., due 3 years hence, at
4| per cent, simple interest ?
106. A and B entered into partnership : A put into stock at first
£2000, and at the end of 8 months £1000 more ; B put in at first £750,
and at the end of 4 months £3000, but took out £1300 at the end of
3 months more. At the year's end they had gained £1635; what should
each receive ?
107. Allowing that 44 J guineas weighed a lb. Troy, when 32 half-
pennies weighed a lb. Av., and observing that a lb. At. contains 7000 gr.
Troy, what was the difference in grains between the weights of a guinea
and half-penny ?
108. How much stock must be bought at 88 per cent., in order that,
by selling out when the stocks are at 90, 20 guineas may be gained ?
1 09. A bankrupt pays 3|(f . in the pound, and the total of his payments
amounts to £154 ; what was his debt?
110. A person has £18752, for which he is receiving 3 J per cent,
but spends annually £27 more than the whole original interest ; what
has he at the end of 3 years ?
111. If £100 be placed at interest at 5 per cent., and the interest be
added to the principal every 20 years, in how many years will it amoimt
to £1000?
112. The prime cost of a 50-gall. cask of wine is £25, and 10 gall,
are lost by leakage ; at what price per gall, must the remainder be sold,
so as to gain 10 per cent, on the whole original cost?
113. To do a certain piece of work A by himself would require 16
hours, B IS, C 20. Suppose that after A and B working together for
5 hours, and then B and C for 3 hours, the remainder of the work 10
left for C to finish, in what time would he finish it?
HISGELLANEOUS EXAMPLES. 138
114. If the carriage of 60 cwt. for 20 miles cost £14^, what can I
have carried 30 miles for £5^ ?
115. Find the side of a square whose area equals 14 sq. ft. 11 in.
116. A and B engage in a speculation, and dividing the proceeds of it,
A took £57 18s., and ££29 148., as their respective portions; what sum
did each lay out, it being known that A paid £7 16s. Sd. more than B?
117. A person had £2950 in the Danish 3 per cents., at 75^, which
he transferred to the Bussian 5 per cents., at 11 Of ; required the altera*
tion in his income.
118. Extract the square root of '009059 and of 464|f, and the cube
root of '578703. •
119. If 7 oxen are worth 64 sheep, and 3 sheep cost £5 128., what
must be given for 100 oxen ?
120. A person buys teas at Ss. and 45. the lb., and mixes them in the
proportion of 4 : 7 ; what will he gain per cent, by selling at 35. 9d, per lb. ?
121. Find the difference between the simple and compound interest
on £150 in 3 years, at 4^ per cent.
122. If 5 men can reap a field, in length 800 ft. and breadth 700 ft.,
in 3^ days of 14 hours each ; in how many days of 12 hours each will 7
men reap a field of 1800 ft. by 960 ft. ?
123. Three soldiers, A^ B, and C, divide 770 cartridges in the follow-
ing manner : as often as A takes 4, B takes 3 ; and as often as A takes
6t C takes 7 : how many will each have ?
124. If £100 in 2 years gain £12 interest, what principal will gain
£6 155. in 4| months?
125. A person desires to exchange 25 Spanish £100 bonds, and
£800i 3J per cent. Stock, for 3 per cent. Consols ; the prices of these
securities being 48, 99, 93| respectively, what quantity of Consols can
he obtain ?
126. A person buys three estates of 56, 67, and 71 acres, and gives
£81 35. 6d., £92 45. Sd., and £109 35. 2d. an acre for them respectively;
what should they produce annually to pay 15 per cent, upon his whole
outlay ?
127. If a beam which is 10 in. wide, 8 in. deep, and 5 ft. 6 in. lorg,
weigh 8 cwt. 1 qr., find the length of another beam, the end of which is
a square foot, which shall weigh a ton.
128. A and B have 185. and 125. respectively; and if A give B
2| + 4| of the difference of 2^-5-13^ of their respective sums, and ^ of
2^ of A' 8 present sum be added to JJ of ^ of Bb, Ca money will be 1 J of
this sum : find it.
129. What is the expense of carpeting a room, 28 ft. long and 19ft.
wide, with carpet | yd. wide, at 55. 9d. a yard ?
134 MISCELLANEOUS EXAMPLE^.
130. A person transfers £2000 stock from the 3 J per cents, at 99,
to the 3 per cents, at 86| ; what is the difference in his income ?
131. Multiply £2 165. 1075^?. by 14433, and divide £9763 14*. 8j«f.
hj 234-5.
132. What would be the purchase-money for an estate producing a
rental of £3228 Zs. 4:d., at the rate of 8| per cent ?
133. What will be the expense of glazing a hall-window contain-
ing 60 squares, each 1 ft. 3 in. long, and llj in. wide, at Is. lOd. per
sq.ft.?
134. A lb. of tea and 4 lbs. of sugar cost 6*. ; but if sugar were to rise
60 per cent., and tea 10 per cent., they would cost 6*. 2d, Bequired
the prices of tea and sugar per lb.
135. If I buy 14 sheep for £39 6s. 5^., and sell 6 of fiiem at 365.
each, for what must the remainder be sold that I may gain 4 per cent
on the whole ?
136. The weights of equal quantities of lead and cork are as 11.324
and .24 ; and 60 cubic inches of lead, with 54 of cork, weigh as much
as 1538| of fir: what number represents proportionally the weight of
fir?
137. By selling an article for 105., the seller loses 6 per cent. ; what
will be the loss or gain when sold for 125. 6^., and what was its prime
cost ?
138. A puts out to interest £2000 at 4 per cent ; he spends annually
£75, and adds the remainder of his dividend to his stock : what is ha
worth at the end of 5 years ?
139. A country containing 711117 inhabitants increases to 732666;
find the increase per cent.
140. If 12 men can complete a piece of work in 15 days, working
6 hrs. a day, how many can do it in 85| days working 12i| hrs. a day ?
141. A bankrupt has good debts to the amount of £456 185. 1«?.,
and the following bad debts, £360 75. 10^., £120 135., and £19 185., for
which he receives respectively 4, 5, and 9 shillings in the £ ; his own
liabilities amounted to £3408 125. : how much can he pay in the £?
142. A had £2 135., and B, when he had paid A 6f-7-S| of
£1 ll5. 6d.f foimd that he had remaining ^ of the sum which A now
had : what had B at first?
143. Find the sq. roots of '0026009 and '0002404, and the cube root
of s
144. A rectangular cistern, of which the length is 13| ft. and the
breadth 6 ft., contains 294^ cubic feet of water ; what is the depth of
the cistern, and what is the weight of water when one cubic inch weighs
252.5 grains?
14 J. At what rate per cent, of simple interest will £1 become a
guinea in 5 years ?
VISCELLAKEOUS EXAMPLES. 135
146. How much will a broker, who charges 5 per cent, discount,
give for a bill for £600 due at 2 months?
147. Eiding a journey of 27 miles into town, I meet the coach which
left town at the same moment that I started from hence (viz. 7 o'clock),
at the 16th mile-stone from town. Supposing that it travels 10 miles an
hour, find the hour when we meet, and the time when (proceeding at
the same rate as before) I shall reach London ?
148. If 12 casks are carried 18 miles for £16 when the carriage is at
Is. Sd.f how fax ought they to be carried for £72 when the carriage is at
lOrf.?
149. Add together f of § of a guinea, | of a pound, and 3^^ of
lis, Sd.; and reduce the sum of l-7-3| of half-a-guinea and 3-»-3| of
1 5s. 6d, to the decimal of a pound.
160. What number of lbs. of tobacco, at the same number of pence
per lb., amounts to £16 105. dd. ?
161. A manufacturer employs 60 men and 35 boys, who work re-
spectively 12 and 8 hours a day during 5 days of the week, and half-
time the other day ; each man receives 6o?., and each boy 2(?., an hour.
What is the whole amount of wages for a year ?
162. A man buys 27 sheep for £30, and sells 12 of them, so that he
loses 3 per cent, in the sale ; at what price per sheep must he sell the
remainder, so that he may gain 2 J per cent, on the whole purchase ?
163. Two persons buy respectively, with the same sums, into the 3
ftnd 3| per cents., and get the same amount of interest ; the 3 per cents,
being at»76, at what are the 3| per cents. ?
154. Find the present worth and discount on £226 Is. lld.t due 7
months hence, at 4| per cent.
155. Three tons of merchandise cost £26 I6s. 6d. ; at how much
per cwt. must it be sold so as to gain 20 per cent. ?
156. Divide 3 J guineas among 6 persons, so that their shares may be
in the prop^ion of the reciprocals of the first 6 units.
157. Divide 999 into three parts, so that 6 times the first, 7 times
the second, and 11 times the third may be equal.
158. Half the trees in an orchard are apple trees, a fourth pear trees,
a sixth plum trees, and there are besides 50 cherry trees ; how many
trees are there altogether ?
159. A banker borrows money at 3| per cent., and pays the interest
at the end of the year : he lends it out at 5 per cent., but receives the
interest half-yearly, and by this means gains £200 a year : how much
does he borrow ?
160. By selling tea at 25. Sd. a pound, a grocer clears |th of his
outlay ; he then raises the price to 35. : what does hft Q\«Mt -^^t <iw:fiu
t^n his outlay at the latter price ?
136 HISCELLAKEOnS EXAMPLESr
161. How much tea, at 2s. i^d., must I give for 28 lbs. of sngar, at
4Jc?., so as to gain 5 per cent, by the exchange ?
162. Eeduce ^^ to its lowest terms, and ^toe^ decimal ; and add
together 2% 3^, ^, and 1| ; and divide 2f of 1| of If by 7||-
163. If 54.32 cub. in. of gold be as heavy as 101.36 cub. in. of silver,
how many oz. of silver are equal in bulk to 226J oz. of gold ?
164. "What is the present worth of £131 12*. 6d., payable in ^ of a
year, at 6 per cent. ?
165. The length of a street is 037 ft. 6 in., and its breadth 66 ft. 8 in.;
find the cost of paving it at 8|ef. per square yard.
166. If 100 men, in 6 diys of 10 hours each, can dig a trench
209 yards long, 3 wide, and 2 deep, in how many days of 8 hours long
will 180 men dig a trench of 360 yards long, 4 wide, and 3 deep ?
167. A person spending annually £240, savas £2 15s. of it quarterly
by ready payment; what is the rate of discount? and if he by this
means makes an increase of 20| per cent, upon his annual saving, what
was his annual income ?
168. A certain sum of money was divided among three persons, A^
Bi C, Suppose that A' a share was £264 12*., and Cs £2 8«., and that
^'s share contained the value of ^s as often as ^s share contained Cs ;
what must the whole amount have been ?
169. Add together 3f of 2i of 7^ of a £, 9? of 3| of a shilling, and
8J of 4| of a penny, and divide the sum by i| of ^ of | of Z^,
170. Extract the square roots of 2'054 and of 4203361 ; and the
cube roots of 15-438249 and 629*422793.
171. If 6000 lbs. of iron are cast off at a foundry in 24 hours, how
many tons weight will be cast oflf in 308 days, supposing them to work
16 hours each day ? and if the price of iron be £3 Ss. per ton, what will
be the gain per cent, upon the annual expenditure, supposing it to be
£20 per week of 6 days ?
172. How must wine, which cost 155. per gall., be sold, so as to gain
21 J per cent. ? and how so as to lose the same?
173. The value of a pound of gold is 14 times that of a pound of
silver, and the weights of equal quantities of gold and silver are in the
ratio of 19 to 10 ; find the value of a bar of silver equal in bulk to £1750
worth of gold.
174. A, B, and C, together, can dig an acre of land in 7J days. A
digs 32 perches in 5 days, and B 54 perches in 7 days. Find the three
lowest integral numbers expresjdng the comparative powers of these men ;
and the time in which C7digs 17| perches.
175. What is the price of a silver cup weighing 1 lb. 10 oz. 12 dwt
6 grs., worth 68. an ounce ?
JZff' Divide tbe cube root of ^^^^^^^ by the square root of 260100.
ItlSCELLAlraiOtJS EXAMPLES. 137
177. Eoduce 2 w. 2 d. 19| hrs. to the fraction of a month, and if of
a shilling + f of half-a-crown + ij of a guinea to the decimal of a £. .
178. A £ist train leaves Bristol for London, a distance of 120 miles,
at 2 o'clock, and travels at the rate of 25 miles per hour ; at what time
must a luggage train, which travels at the rate of 15 miles in 50 minutes,
have left, so as not to be overtaken by the fast train ?
179. Find the commission on £126 at f per cent., and reduce the
answer to the decimal of £l lis. 6d.
180. If, by selling fine Irish cloth at 5s. per yard, I gain 8 per cent,
what will be my rate of profit if I sell at 6«. 4e?. per ell ?
181. Add together the cube roots of '007301384 and 32768, and
multiply the result by the square root of 72J.
182. What ready money will discharge a debt of £528 ds., due
4 months hence, at 4| per cent. ?
183. Find the least common multiple of 64,720,960 ; and find what
decimal 17 yds. 1 ft. 6 in. is of a mile, and what fraction of 3«. 6d. is |
ofi|of2«. 6(;.?
184. The 3 per cent, stock is at 98|, and the 3 J per cents, at 106| ;
into which is it most advantageous to buy ?
185. £1000 is to be divided among Aj B, and C7, so that for every £3
given to ^, .8 is to receive £5 and C £8 ; what sum had they each ?
186. Reduce 4^ lbs. Av. to Troy weight, and 3 cwt. 34 lbs. 2 oz.
to the decimal of a ton ; and '0975, '63, *524d, to their equivalent frac*
^ons.
187. The quantity of copper ore sold at Truro on a certain day was
3696 tons (of 21 cwt. each), and the produce 6| per cent. ; find the quan-
tity of fine copper obtained from it in common weight.
188. A rectangular parish, 6 fur. long and 4 fur. broad, is enclosed ;
a belt of plantation, 200 ft. wide, is carried the whole way round ; a
main road, 60 ft. wide, runs across the land in the direction of its length,
and a cross road, 41 ft. wide, in the direction of its breadth : how many
acres of field were there ?
189. If the sixpenny loaf weigh 5| lbs. when wheat is at 5|«. per
l)U8hel, what must be paid for 52^ lbs. of bread when wheat is at 8^. 6d.
per bushel ?
190. Find the present value of £273 Os. 9d., due 3 months hence, at
4^ per cent., and the compound interest on £105 in 3 years, at 3| per
cent.
191. A body of 7300 troops is formed of four battalions, so that | of
the first, § of the second, | of the third, and f of the fourth, are all com-
posed of the same number of men ; how many were there in each ?
192. Among the Jews the coin mina (or pound) was wotOx §v<^
shekels of silver, each weighing 219 grs. ; the tueigM TD^Tia^-sR\i^xi^'5. ^^^
138 MtSCCLLANEOtJS EXAMPLES.
weighed 100 shekels, when of silver, 60 ; what were the values of these
minse, rating gold at £4 and silver at 55. an ounce ?
193. A father left to the elder of his two sons ^ of his estate, and||
of the remainder to the yoJinger, and the residue to his widow ; find
their respective legacies, it being found that the elder son received
£1690 more than the younger.
194. Divide 240 into two parts, such that \ of one added to ^ of the
other shall equal 36.
195. If 193 Russian versts be equal to 205*0 French kilometres, and
1552'94 kilometres equal to 964'9 English miles, how many miles are
equal to 100 versts?
196. If the rent of 2 acres for | of a year be £l 3a. 3<?., what will be
the rent of 547 acres for a half year?
197. If I buy 3 per cents, at 78|, and 3J at 95^, which is the best
investment? If I had invested £6962 Ids. 3^. in each, and the former
rose and the latter fell ^, how much should I lose or gain ?
198. If 3 men can mow 7 acres of grass in 5 days of 9 hours each,
in how many days of 8 hours each will 5 men mow I75 acres ?
199. Add together 3^, 2{]^, J, and ^ ; find the difference of 3^^ and
2|, and divide 3^ by 2^.
200. Five thousand copies are issued of a 65. book: the cost of
printing is Is. per copy, of binding 4c^., and of carriage, advertising,
&c., 2d, : the publisher disposes of them to the retail bookseller, charg-
ing 25 copies as 24, and 30 per cent, less than the selling price, and
upon the whole receipts takes 10 per cent, commission for himself: what
are the gains respectively of author, publisher, and bookseller on this
edition ?
201. Find the square root of iff, and the cube root of 352045*367981.
202. Find the discount on £1294 105. for IJ year, and the interest
on the discount for the same time, at 4| per cent.
203. Divide 100 guineas into an equal number of guineas, half-
guineas, crowns, half-crowns, shillings, and sixpences, and reduce the
remainder to a fraction of a pound.
204. A person has £3500 to lay out ; the 3 per cents, are at 82|, and
the 3 J at 96 : what would be his income from each ?
205. How many inches are there in the diagonal of a cubical foot,
and how many square inches in a superficies made by a plane passing
obliquely through two opposite edges ?
206. A merchant employs £700 in trade, and at the end of 3 years
takes another into partnership, who advances £1900. At the end of 4
years from this time they have gained £500; how ought this to be
divided between them ?
207. If 24 pioneers, in 2| days of 12| hours long, can dig a trench
MISCELLANEOUS EXAMPLES. 139
f 3975 yds. long, 4i yds. wide, and 2^ yds. deep, what length of trench
will 90 pioneers dig in 4^ days of 9| hours long, the trench being 4|yds.
wide and 3} yds. deep ?
208. What is the discount on £257 Ss. 8|<f., paid 210 days before
due, at 4| per cent. ?
209. What is the cost of papering a room 15 ft. long, 12 ft. wide,
and 10 ft. high, with paper 30 in. broad, at 7J«^. per yard ?
210. The sum of £925 was so divided among A, B, Q and D, that
B'a portion was equal to ^ of A% Cb was equal to | of E's, and i>'s was
hftlf as much as j^s and C"s together : what did each receive ?
21 1. A draper bought 5 pieces of silk, each 52 yards, at is, 3^. per
yard, and sold the whole so as to gain as much as 16^ yards were sold
for ; what was the selling price per yard ?
212. £100 stock, in the 3 per cents., is sold for £91 15^. ; how much
can be bought for £540, allowing, for commission, | per cent, upon the
stock bought?
213. A gentleman's income is £896 135. id. per ann. ; he gives to
the poor quarterly £13 10^., and lays up 200 guineas at the year's end :
how much does he spend in 6 days ?
214. A grocer buys 13 lbs. of tea at 25. 3c?., 16 lbs. at 25. 5d., and
18 lbs. at 35. Zd.f and mixes them : at what rate per lb. must he sell the
mixture so as to gain on the whole 17| per cent. ?
215. What is the present worth of £2035 15s., due in 2 yrs. 5| mo.,
at 4| per cent. ?
216. What is the expense of paving a rectangular court-yard, whose
length is 63 ft., and breadth 45 ft., it being paved with pebbles at l5. dd,
per sq. yard, except a foot-path, which runs the whole length, 5 ft. 3 in.
broad, and is paved with flag-stones at 35. per square yard ?
217. A and B can do a piece of work alone in 12 and 16 days
respectively ; they work together at it for 3 days, when A leaves it, but
B continues, and after 2 days is joined by C, and they finish it together
in 3 days ; in what time would C do it alone ?
218. Find the value of 13^ of 2 cwt. 2 qrs. ; and of ff of £8 85.
&^.
219. A can mow 2^ acres of grass in 6§ hours, and B 2| acres in 5|
hours : they mow together a field of 10 acres ; in what time will they do
it^ and how many acres will each mow ?
220. In making gold thread for embroidery, a cylinder of silver
weighing 360 oz. Av. is cased with one of gold weighing 6 oz. ; and this
mass is drawn through a series of circular holes, continually diminishing
in diameter, until it becomes so thin that 202 feet in length weigh one
dram : what is now the length of the thread ?
221. There is a circular field, a mile Ia cixcaxatei^\i!^^^ «sAli«'M.^
140 MlSCSLLANEOtS EXAMPLES.
and A" start together at one point in ihe circumference, to Udvel r.und
it in the same direction, at the respective rates of 84, 104, and 144
yards per minute. How soon will the three be again together?
222. The sum of four fractions is 2^|, and one common result is
obtained by adding the fraction ^^ to the first, subtracting | from the
second, multiplying the third by |, and dividing the fourth by JJ.
Find the four fractions.
223. If the duty on two kinds of foreign wine be changed from
3«. 6d. and 28., respectively, per gallon, to a uniform rate of 28. 6d.,
and, in consequence, the consumption of the former, which was at the
rate of 200,000 givllons per annum, is increased 15 per cent., and that
of the latter, which was 700,000 gallons, is diminished 25 per cent.,
while the cost of collecting the duty is reduced £rom 2d, to l^d. per
gallon ; find the gain or loss to the revenue.
22 1. A contractor undertook to supply coals at 22^. 6d. per ton.
He delivered them by weight in 8acX:8, each sack estimated to weigh
3}lb., and to contain 1^ cwt. of coals. If the sacks really weighed 71b.
each, what was the money value of his fraud on 1000 tons charged*
and what was the increase per cent in the price of the coals?
141
APPENDIX I.
REMARKS ON THE TABLES.
The choice of the number 10, as the base or radix^ as it is called,
upon which the decimal system or scale of Notation depends,
common as it is to so many nations, l3arbarous as well as civilised,
may be conceived to have had its foundation in the natural practice
of counting on the fingers, whence the term digit; but we might
have taken any other number for base, and, having characters for
zero and all the figures less than the base, we might express any
number whatsoever in such a scale. (^See Alg, Notation.)
The admirable method of notation by the use of the vine digits
and zero is of extreme antiquity ; and though called the Arahic
method, (because first introduced into Europe through the Moors
in Spain about the 11th century, though it was not till about
the 14th that it superseded the old Roman system,) was certainly
known to the Hindoos long before the rise of Arabian science,
and even by them ascribed, for its excellence and the remoteness
of its origin, to the direct revelation of the Divine Being. It
seems to have been traced with some probability to the regions
of Thibet.
The system of the Greeks was almost identical with that of the
Hebrews, or Phoenicians : that of the Romans, though very simple,
was singularly cumbrous and inconvenient; and it is a striking
proof of their extreme indifference to any advances in scientific
matters, that they so pertinaciously retained it, notwithstanding
their acquaintance with the far more perfect and comprehensive
notation of the Greeks.
The ^gures now in use are derived from the old Arabic, though
much modified and corrupted by the course of time.
When numbers are used with reference to the things numbered,
as when we say 3 apples, 4 penSf 6 shillings, they are said \» \si^
L
142 APPENDIX I.
concrete numbers ; when used without such reference, merely to
indicate a certain number of units of the same kind, as when we
say simply 3, 4, 5, they are called abstract numbers.
The concrete quantities, required in ordinary calculations, are
those which are necessary to express Money ^ Weighty Space^ and
Time, In the Tables will be found the most common of these
quantities ; but we shall here make a few additional remarks about
them, and explain the Standards^ which are used in each of these
classes.
The standard gold coin of this realm is made of a metal, <^
which 22 parts in 24 are pure gold, and 2 parts alloy, a mixture
of silver and copper. From a lb. Troy of this metal are coined
46^§ sovereigns = £46 14*. 6d. ; so that the Mmt price per oz. of
standard gold = -jJy of £46 14*. 6d. = £S 17 s. lO^d. ; and since there
are 11 oz. of pure gold in 12 oz. of standard, we shall have (neg-
lecting the vsdue of the alloy) the value per oz. of pure gold at the
Mint = T'T of £46 14*. 6d = £4 4s, ll^d.
The standard silver coin is made of a metal, of which 37 parts
in 40 are pure silver, and 3 parts alloy (copper). From a lb. Troy
of this metal are coined 66s., so that the Mint price per oz. of
standard silver is 5s. 6d. : and since there are ^^ of an oz. of pure
silver in this, the value per oz. of pure silver at the Mint is H
of5s.6d. = 5s.lli^d.
From a lb. Av. of copper are coined 24 pence : but this is not
a legal tender for more than 12c?., nor is the silver coinage for more
than 40s., the gold coinage being the standard of the realm.
The following coins are noticeable, occurring oflen in ancient
documents : —
f Groat = 4d., Tester = 6d., Noble = 6*. Sd,, Angel = 10«.,
Merk=13i;. 4d., Carolus = 23*., Jacobus = 25^., Moidore =
27«.
Great inconvenience having been long felt in this country, from
the want of uniformity in the systems of weights and measures,
which were in use in different parts of it, an Act of Parliament was
passed in 1824, and came into operation on Jan. 1, 1826, by which
certain weights and measures, therein specified, were declared to
be the only lawful ones in this realm, under the title of Imperial
Weights and Measures,
BEMAEKS ON THE TABLES. 143
It was settled by this Act —
1. That a certain yard measure made by an order of Parliament
in 1760, (by comparison with the yards then in common use,)
should be henceforward the Imperial Yard, and the Standard of
Length for the kingdom : and that in case this Standard should be
lost or injured, it might be recovered from the knowledge of the
fact, that the length of a pendulum, oscillating in a second, in
vacuo, in the latitude of London, and at the level of the sea,
(which can always be accurately obtained by certain scientific
processes,) was 39' 13929 inches (or thirty sixth -paxtsi) of this jaxd;
2. That the half of a double-pound Troy, made at the same
time, should be the Imperial Pound Troy, and the Standard of
Weight ; and that of the 5760 grains, which this lb. contains, the
lb. Ay. should contain 7000 : and that in case this Standard should
be lost or injured, it might be recovered from the knowledge of
the fact, that a cubic inch of distilled water, at the temperature
of 62** Fahrenheit, and when the barometer is at 30°, weighs
252.458 grains ;
3. That the Imperial Gallon, and Standard of Capacity, should
contain 277*274 cubic inches, (the inch being above defined,)
which size was selected from its being nearly that of the gallons
already in use, and from the fact that 10 lbs. A v. of distilled
water, weighed in air, at a temperature of 62®, and when the
barometer is at 30% will just fill this space.
The name Troy Weight has been derived from Troyes, a city
of France, where great fairs were once held, and to which it was
introduced, about the time of the Crusades, from Cairo in Egypt ;
but it has also been derived from the monkish name for London,
Troynovant, from Trinovantum. The name Avoirdupois is probably
derived from the old Norman, avoirs, goods and chattels, smdpois,
weight.
It is probable that a grain of wheat was the element of weight
in former dayg, and a grain of barley (barleycorn) the element of
length*
The pennyweight was so called as being the weight of the silver
penny tiien in use.
The words ounce and inch are both derived from the Latin
uncia^ or twelfth part, of a pound and foot respectively.
l2
144
APPENDIX I.
The following weights and measures are noticeable, besides
those given in the Tables.
Carat (of Diamond) . =3| grs.
Carat (of Gold or Silver) =240 grs.
Firkin (of Butter) . .
Fodder (of Lead) . .
Great Pound (of SUk)
Pack (of Wool) . .
Yard (of Land) . .
Hide (of Land) . .
Firkin (of Beer)
Kilderkin
Barrel . . .
Hogshead .
Butt . . .
Tun . . .
» 56 lbs.
=sl9icwt.
e24 oz.
«= 240 lbs.
s 30 acres
= 100 acres
Line • .
Barleycorn
Span . •
Cubit . .
Pace . •
Degree •
Flemish £11
French Kll
«iiiL
=Jin.
=9 in.
= 18 in.
-5ft.
= 69| miles
= 3 qrs.
= 6 qrs.
= 9 gals.
= 18 gals.
=36 gals.
«a 54 gals.
= 108 gals.
= 2 butts
Anker (Wine or Spirits) = 10 gals.
Bunlet =18 gals.
Tierce =42 gals.
Hogshead . . . . = 63 gals.
Puncheon •» 2 Tierces . = 84 gals.
Pipe =2 Hogsheads . =126 gals.
Since there are 24 carats in a lb. of gold, the fineness of gold is
often expressed by saying that it is so many carats fne^ meaning
so many parts out of 24 ; thus our standard gold is 22 carats fine,
and jewellers' gold (as marked on the stamp of a watch) is 18 carats
fine.
In measuring land, surveyors use a cAatn, called Gunter*s chain,
which is 22 yards long, and divided into 100 links; and 10 square
chains, or 100,000 square links, make an acre.
The Greek unit of linear measure was the itovq = 12 '1 35 inches.
The principal Attic measures of length were
hoLKTvKos (iir) = I in. nearly. irKiepov ( 1 OOir) = 1 1 1 f^
irfix»s (l|^) = l|ft. or|yd. ariJbiov (600ir) = 606f ft.
dpyvla (6ir) = 6 ft. or a fathom. hiavXos (12 OOir) = 1 2 1 3i ft.
It will be found that there are very nearly 8 J stadia in a mile.
The Persian parasang was 30 stadia, rather more than a league.
The principal square measures were the square ttovq and nkkPpov,
which latter contained 4 dpovpai, and was a little less than a rood*
The Bomap unit of length was the ;7C5 = 1 1*6456 inches.
BEMABKS ON THE TABLES.
145
Their other ordinary measures were the digitus (^ pes)^ uncia
(^p,)ypalmu8 (^p.%palmipes (lJ/>.)» cubitus (lip.), gradus (2lp.),
passus (5j5.), milliarium or mille passuum (5000 p, = 1618 yds.).
Their principal square measure was the jugerum (240/). by 120)
s= 28800 pedes quadratic or f acre, nearly.
For rough calculations, the ttovs and pes may each be considered
to be equivalent to B.foot English.
The Greek and Roman systems of money were naturally founded
upon those of weight, the denominations of money and weight
being identical.
The Attic unit of weight and money was the drachma, which,
as a weight, was equivalent to 66^ grs. ; and this weight of silver
being worth 9}c?., this was the value of the silver coin, drachma.
Their other coins (all in silver) were as follows —
6 obols (6fio\o() made 1 drachma (^paxfi'f})
100 drachma ... 1 mina (fiya)
60 mina .... 1 talent (rd\atnov) ;
80 that the obol was worth about 1^., the mina £4 Is. Sd., the
talent £24S 15s.
Besides these, there were the dioholus, triobolus, didrachm, tetro'
drachm (or stater), &c., whose values are explained by their names.
In later times, the value of the drachma as a coin corresponded
to the Roman denarius s= S^d,
The Roman unit of weight was the libra, or pound, = 5204 grs.,
tiiat is, nearly ^ lb. Av., or very nearly -^ lb. Troy. This weight
of the metal as or bronze (a mixture of copper and tin) formed
originally the coin as, or pound; but the weight of the coin was
subsequently reduced in the proportion of 8 : 5.
The as or libra was divided into 12 uncia, i. e. twelfth-parts ,
and the following names were given to the different multiples of
an uncia.
1 1 tine, (sesqui-oncia) sescunx
2 ... (Jib.) sextans
8 ... (\\b.) tet'uncius or quadrans
4 ... (I lb.) triens
6 quincunx
t ... (^ lb. ^semi'Os) semis
7 unc
8
9
10
11
12
septunx
(I lb.) bes
(I lb.) dodrans
(fib.) dextans
deunx
Zt^raoxoa
146 APPENDIX I.
The name bes is supposed to be formed from des (as bis from ^I'c),
and this from de-triens (desit triens\ meaning an as wanting a trieia
or third ; just as dodrans, dextans, deunx, are formed from de^quad*
rans, de'Sextans, de-uncia.
It should be observed that the word undo, or ounce, means
simply a twelfth-part; and therefore the above terms sescunx,
sextans, &c. were used by the Romans, as so many fractions, for
subdivisions of other units, as well as of the as : thus, we have had
above the uncia of length =-^j9e«, and see also below among the
measures of capacity.
The uncia of weight = 434 grs. = very nearly an ounce Av.
The Romans had also a silver coinage, consisting of the denarius
and its parts. These were the denarius, worth 8 Jc?., and equivalent
(as its name denotes) to' 10 ancient ases or 16 later ones ; the qui"
narius (5 ancient ases) = 4^d,, called also victoriaius, from the image
of Victory upon it ; the sestertius (i. e. semis tertius nummus, or
a coin worth 2^, viz. ancient ases)=^2ld,; libella^^^^den., sem"
hella (semi-libeUa) = ^ den,, t€runcius^= t^ den. = (as above) J an-
cient 05 or ^ later as^\d., nearly.
For rough calculations we may reckon the as at ^., sestertius 2d,,
denarius S^d, The sum of 1000 sestertii was called a sestertium
'= £8 I7s^ Id., but there was no coin for this amount.
The Greek ?f (rr^c = Roman sextarius. may be conveniently taken
as the unit of capacity, being equivalent to ('9911 or) just one pint
English. The sextarius was so called as beii^ ^ of the congius, and
contained 12 Kvadot, cyathi ; and the multiples of the cyathus had
the same names among the Romans as those of the uncia, or ounce
of weight : thus, 2 cyathi was a sextans, or ^ of a sextarius, &c.
The Greeks had also the icorwXiy = J sext,=^ pt., ;^oTi/t$= 1^ pt,
xovff = 4 xo*vtf«ff = 3 qts., /ifrf)ijriyj:=9 gals., fisdifivoQ=: 12 gals.; and
the €KTos and rffiUKrog were the sixth and twelfth parts of the mC'
dimnus. The Romans had, beside the cyathus and sextarius, the
hemina = ^ pt., congius = 6 sext* = 3 qts., modius = 2 gals., uma^
3 gals., amphora = 6 gals.
A Solar Day is the interval between two successive transits
of the Sun over the meridian of any place ; but, from several
causes, this interval is continually varying, though slightly, id
BBMABKS OK THB TABLES. 147
€(uration. If, however, we take the mean of many observations,
we shall get the length of the Mean Solar Day^ and this is the
Standard unit for the measurement of Time in ordinary life;
though Astronomers have another unit in common use.
The Solar Year is the interval between the Sun's leaving and
returning to a certain fixed point in his apparent orbit round the
Earth (the Ecliptic)^ and is accurately determined by Astronomers
to contain 365.242218 mean solar days = 365 days, 5 hrs., 48 min.,
47 J sec. nearly. Hence the common, or Civile Year, which contains
only 365 days, is somewhat shorter than the Solar, or True, Year ;
and this error, being nearly J of a day, would accumulate, if not
corrected, so as to produce at length a complete confusion in the
times at which the seasons would return, and we should have
Summer, sometimes in July, sometimes in December.
Julius C<B8ar first corrected this ; and, supposing, in the then
Btiate of Science, that the Solar Year contained exactly 365 days,
6 hrs. = 365*25 days, he ordered that every fourth year should
contain 366 days instead of 365. But this correction was really
too great by '007782 of a day, since the Solar Year contained
only 365*242218 days; and in 400 years this error amounted to
400 X •007782 = 3*1128 days; and hence it happened that the
yemal equinox, which fell, in a.d. 325, at the Council of Nice,
on March 21, fell in a.d. 1582 on March 11. Pope Gregory, in
consequence, caused 10 days to be omitted in that year, making
Oct. 15 to follow Oct. 4, so that the vernal equinox fell next year
again on March 21 ; and, to prevent the recurrence of this error,
lie ordered that in every succeeding cycle of 400 years, 3 of the
leap years should be omitted, viz. those which complete a cen-
tury, when the number of hundreds is not divisible by 4 ; thus,
1600, 2000 are leap years, but not 1700, 1800, 1900, &c.
The Gregorian correction was introduced in England in 1752,
when it had become necessary to omit 11 days of the current year;
and the Calendar thus rectified is called the New Style, the Julian
reckoning (which is still retained in Kussia) being the Old Style.
This correction is too great on the other side by *000282 of a
day, but the error only amounts to a day in 4000 years.
N.B. — Until A.D. 1752, the New Year's day in England for all
official records was the 25th March : hence, we often find, in works
148 APPENDIX I.
relating to an earlier period, a double date given, as 1703-4,
whenever the event referred to occurred during the month of
January, February, or March, up to March 25 — ^the former indi-
cating the year according to the old, and the latter, according to
the modem, reckoning.
149
APPENDIX IL
DECIMAL COINAGE.
1. It maybe desirable to say here a few words upon the
subject of a Decimal Coinage, which has been for some time
under the consideration of the Government, has been recom-
mended for adoption by a Committee of the House of Com-
mons, and is likely, therefore, before long, to be introduced
in England, as it has been already in France and in the
United States of America.
2. Two systems of decimal coinage have been proposed,
and each has met with warm supporters, — the one based
upon the penny or farthingy the chief coin of the poorer
classes, as the unit of reference, the other upon the pound
sterling or sovereign, the chief coin of the wealthier classes.
Each of these systems has its own peculiar advantages and
disadvantages, which we shall proceed briefly to explain.
Of the two, the advantages of the latter, based upon the
pound sterling, seem to be upon the whole the greatest;
and as it has been specially recommended by the House of
Commons' Committee, it is probably that which will be
ultimately sanctioned by Act of Parliament, perhaps, with
some modification of its details, as, for instance, in the
names at present proposed for the new coins.
3. L One system of decimal coinage takes the farthing
for its unit of reference, and its money-table would be some-
what as follows : —
10 Farthings make 1 Doit = 10/: = 2|J.
10 Doits make 1 Florin =' 100/ = 2s. let.
10 Florins make 1 Pouncf =100qf.=^^Qs. \C^
150 APPENDIX II.
The coins required for use in this system would be the
following : —
Copper — farthing, halfpenny, and pennj, as now ;
Silver —doit (2j</.)» S^oat (bd.), shilling (\^d.),Jbrin (25d,) ;
Gold — half-pound (1 25rf.)» pound (250flf.)'
It might also be convenient to have a dollar or double*
florin (50d.) in silver, and a crown (62^d.) in gold, so that
five dollars, or four crowns, would go to make the pound.
The difference in size between the doit and the groat, being
much greater than that existing between the present Sd,
and 4d. pieces would allow very well of their being both
coined in silver.
4. The advantages of this system are the following : —
(1.) All coins now in use would be still available; and
thus, while the banks would be collecting the old coins,
and gradually withdrawing them from circulation, business
might be carried on as usual with the old shilling, florin,
and pound. This would prevent, no doubt, much confusion
at first, especially among the poorer classes.
(2.) The farthing, halfpenny, and penny, would be per-
manently retained, and the price of food, the rate of wages^
&c., being generally fixed by the penny, much inconve-
nience would be saved by this means to the mass of the
population.
(3.) No change need be made in the penny postage,
the penny-stamp, the tolls for turnpikes, bridges, &c., noi
in any fixed payment whatever, as now existing.
5. The disadvantages of this system are the following:' —
(1.) The present pound sterling, which is the usual unit
of reference in all great questions of national and com-
mercial finance, would be ultimately displaced altogether.
(2.) The accounts of bankers, merchants, &c., kept during
past years according to the old coinage, or the sums of
money mentioned in statistical or other records could not
be immediately compared with corresponding entries under
^Jjo new sjratem, nor without the trouble of reducing them
DECIMAL COINAGE. 151
in each case to their equivalent expressions in the new
coinage.
(3.) The process of reduction from the old coinage to the
new, though easy on this system, is much more easy on
the other system of decimal coinage, as will presently
appear.
6. "We may here complete what we have to say on this
system, by explaining the process of reduction from the old
coinage into the new.
To reduce a Sum of Money from the present Coinage into
the new Decimal Coinage {Penny System).
Since one old pound contains 960 farthings,
and one new pound contains 1000 farthings,
it follows that if a denote the number of old pounds, and
b the number of new pounds, in the same given sum of
money, then
960a= 10006*, or 6=^ a=(l-i^) a=a-.04a.
Hence we may find the number {b) of new pounds, cor-
responding to any given number (a) of old pounds, by sub-
tracting from a the quantity '040, which we obtain by
merely multiplying a by 4, and moving the decimal point in
the result two places to the left, or otherwise by deducting
4 per cent, from the amount.
Ex. 1. Beducc £765 from the old Coinage into the new (Penny)
Coinage.
Here a « 765*00
•04a = 30-60
b = 734-40 = 734 Pounds 4 Florins (new Coinage).
Since 1 shilling = (^V = T^ir = ) '05 of a pound, any
number of shillings in the given sum may be expressed at
once as a decimal of a pound, by merely multiplying by 5,
* For if we denote that sum by S when reduced into farthings,
S S
960'='''*"'^ 1000 = *'
5=9600 = 10005.
• •
152 APPENDIX 11.
and setting the product to fill the two places of figures
immediately after the point
£x. 2. Reduce £343 17«. into the new (Penny) Coinage.
Here a ^ 343*850
•04a = 13754
b := 339096 => 330 Pounds, Florins, 9 Doits, 6 /.
(new Coinage).
If there are any odd pence in the given sum, these have
only to be reduced to farthings, and added in as thousandths
of a pound.
£x. 3. Redace £409 Us, 8|</. into the new (Penny) Coinage.
Here a = 409'550
•04a = 16-382
393168
8|rf. = 34
b = 393-202 = 393 Poitnds, 2 Fl, 2/. (new Coinage).
7. The converse process of reduction from the new
coinage into the old would be performed as usual.
Ex. 1. £734-4 (new) = 4) 734400/
12 )183600 rf.
20) 15300* .
765£ (old).
Ex. 2. £330*096 (new) = 4 )330096/
12)82524rf.
20)6877*.
343£ 17 s, (old),
Ex. 3. £393-202 Cnew) = 4) 393202/:
12 )98300^ .
20) 819U . 8l<f.
409£ Us. Sjd (old).
8. II. The other system of decimal coinage takes the
pound sterling, or sovereign, for its unit of reference, and
its money-table would be somewhat as follows: — the mil
being the -^^^ of a pound sterling ^^ of a penny = |f of
a farthing.
10 Mils make 1 Cent « ^£ = 2fd:
10 Cents make 1 Florin « ^ £ « 2».
10 Florins make 1 Pound sterling » 20s,
DECIMAL COINAGE. 153
The coins required for use in this system would be tiie
following : —
Copper — mil (^.)> tuHhtniU or double {^.), five-mils or doit
Silver — c^t (2f<i), two-cents or groat (4|c/.), five-cents or
shiUing (\2d.), florin (2».).
Gold — half-sovereign (10«.), sovereign (20«.).
It might also be convenient to have a dollar or double*
florin (45.) in silver, and a crown (55.) in silver or gold.
9. The disadvantages of the system are the following : —
(1.) It would abolish the coins most in use with the poor,
namely, the farthing, halfpenny, penny, and 3c?., 4ef., and 6<i.
pieces, leaving them only the shilling, and coins of larger
value. The sixpence^ indeed, might still be used for a time,
as it is exactly equivalent to 2S mils ; but it would ulti-
mately be withdrawn from circulation.
(2.) It would be impossible to pay exactly in the new
coinage a sum in the old coinage which contained (besides
pounds and shillings) any number of pence, except it were
Ma?-pence. For lef. =4^ mils, 2d, = 8^ mils, &c.
(3.) Hence also it would be necessary that, wherever a
rate o£ Id. is now levied for any purpose, a change should
be made, and either 4 mils or 5 mils charged instead.
Where large sums are raised by such a rate, this would
produce a very considerable difference in the amount so
obtained.
To take, for instance, the case of the penny postage : if 4
mils be charged instead of Id, = 4^ mils, the loss to the
government upon every penny would be ^ mil, and upon a
million of pounds 240000000 x ^ mils = 40,000,000 mils =
j640,000; whereas, if 5 mils be charged instead of Id,, the
gain to the government would be ^ mil upon every penny,
or, upon a million of pounds, ^200,000.
' The same would be true of tolls taken for turnpikes,
bridges, &c., which are usually rated at Id,, 2d,, Sd,, Ad.,^
&c., and the difficulty of coming to a. EaVV^^aieVw^ ^\xiaccv^^-
154 APPENDIX II.
ment in sucli cases would be mucli greater than in that of a
government impost. For, in the latter case, it is tbe
government, that is, the nation itself, which would be the
gainer or loser by the loss or gain of the public in paying
the tax; whereas, in the former, the loss or gain of the
public would occasion a corresponding gain or loss to the
private individuals or companies who might be the pro-
prietors of the tolls.
10. Notwithstanding the above disadvantages, the re-
commendations of this system are so great, (1) from its not
abolishing the shilling^ florin^ crovm, half-sovereign, and
sovereign; (2) from its allowing old accounts to be com-
pared at sight with those of the present day, without the
trouble of reduction ; (3) from the facility with which a
sum may be converted on this system from the old coinage
into the new ; that there is little reason to doubt its being
ultimately adopted, if our present system is exchanged for
any other.
11. To reduce a Sum of Money from the present Coinage
into the new Decimal Coinage {Pound System).
Here the number of pounds remains unchanged; the
shillings, if any, may (as before) be expressed as a decimal
of a pound by multiplying by ^^ ^^ '^^ 5 an<J> since l</.=4^
mils, if the pence be converted into farthings, the number
of farthings will give the number of equivalent mils, except
that 1 mil must be added whenever the number of pence is
6d.y or above it. If special accuracy be required, then 1 mil
should be added for any number of odd pence between 3rf.
and 9d,y and 2 mils for any number of odd pence above 9d. ;
by which arrangement the loss and gain upon the fractional
parts of a mil, when there are several sums of money con-
cerned, would in the long run be fairly balanced.
Ex. Reduce £409 Il«. 8^. from the old Coinage into tbe new
(Pound) CJoinage.
Here £409 1I«. Od,^£409'5bO
8jc/. 35
Ans. £409*585 =£409 5fl. (85 cents, or) 8 centi
5 m\\&.
DECIMAL COINAGE.
155
12. The converse operation would be performed as
usual.
£409*585
20
11-700
12
8*40
4
1-60
Ans, £AQ9 \\8, S|<f. nearly.
13. We may exemplifj the application of this system in
one or two instances.
Ex. 1. Multiply £37 17«. 4|rf. by 43.
Old Coinage,
New Coinage.
£37 17 4}
£37-850
10
19
£378 13
9
4
£37 17 4|
£1514
t: ^
aa
37-869
43
113607
151476
£1628 7 li = £1628356 (new coinage). £1628-367
N.B. — The difierence in these two results arises from the fact that in
the one we have expressed l|(/. by 6 mils, instead of 6| mils, its true
Talae, and in the other we have expressed 4|(/. by 19 mils, instead of
18| mils, its true value. The second error of \ mil when multiplied
by 43 produces an error of 10| mils, which added to the first error of \
mil makes up the whole difference of 1 1 mils.
£x. 2. Find the value of 5 cwt. 3 qrs. 14 lbs. at £14 9«. 8(f. per cwt
£14
9 8
8=
£14-483
5
72*415
2 qrs.
1
o
2
7-2415
1 qr.
3-62075
14 lbs.
1-810375
£85-087625 « £85-088 (nearly) = £85 Ofl, 88/
14. It would be of little use to pursue this subject any
further at present, while the whole matter is yet under
consideration, and the details of the measure, to be her&«
after proposed to Parliament, are by no means fixed.
156
APPENDIX IIL
THE METRIC SYSTEM,
15. Besides the Decimal Coinage, there is also a Decimal
System of Weights and Measures, commonly called the
French or Metric System, which has been adopted by nearly
all the Continental nations of Western Europe,* and will
probably at no very distant day be established also in
England. The first step indeed to such establishment had
been already taken, when the Council of Education required
in their Code of Regulations (1871) t that a chart of the
Metric System should be hung conspicuously on the walls of
all schools under Government inspection, and that in all
such schools children in Standards V and VI should know
the principles of the Metric System, and be able to explain
the advantages to be gained from the uniformity in the
method of forming multiples and sub-multiples of the unit. J
* The Metric System has been adopted in France, Holland, Belgium,
Greece, Spain, Portugal, Italy, Boumania, the North Gennan Confede-
ration, Wurtemberg, Bavaria, Baden, and also by Chili, Equador,
Uruguay, Brazil, the Argentine Confederation, New Granada, Peru,
Venezuela, and partially or in substance in Norway, Canada, British
India, and the United States ; while a Decimal System of Weights and
Measures, differing only from the Metric System in the unit chosen as
the base of the System, exists by law in Austria and Switzerland.
t But this rule is not at present (1874) in force.
X In 1864 the Metric Act of Parliament (27 & 28 Vict. c. 117) was
passed, which provides that, * Notwithstanding anything contained in
any Act of Parliament to the contrary, no contract or dealing shall be
deemed to be invalid or open to objection on the ground that the weights
or measures expressed or referred to in such contract or dealing are
weights or measures of the Metric System, or on the ground that
decimal subdivisions of legal weights and measures, whether Metric or
otherwise, are used in such contract or dealing.* In other words, this
Act permitted the use of the Metric System. And yet, * by a strange
THE METEIC SYSTEM. 157
16. The advantages in question are obvious. Thus in
Avoirdupois Weight 16 drams make 1 ounce, 16 ounces
make 1 pound, 28 pounds make 1 quarter, 4 quarters make
1 hundred- weight, 20 hundred-weight make 1 ton, where
the numbers, indicating the multiples of the unit of the
next lower denomination which make one of the higher, are
respectivelj 16, 16, 28, 4, 20 ; and so in Troy Weight thoy
are 24, 20, 12, in Apothecaries' Weight, 20, 3, 8, 12 ; and
the same irregularity prevails in the Tables of Measures.
But in the Metric System the number is always the same,
viz, 10, so that ten times the uniii 9f the next lower denomina-
tion makes always one of the higher — except a slight modi-
fication in Square Measure, as shown below. By this means
all laborious multiplications and divisions are avoided, such
as are required under the old system, e,g. for reducing
ounces to tons, or miles to inches. And arithmetical opera-
tions of all kinds are so much simplified in practice by the
use of the Metric System that (to use the words of Prof.
Leoke Levi, Metric System, p. vi), * Here is a tool which
offers facilities for saving one-half of the time in arithmeti-
cal education, and one-fourth, or one-third, of the time spent
in all the transactions which include calculations of weights
and measures.' Being, moreover, so generally employed
on the Continent, it is very desirable, with a view to inter-
national communication, that it should be as soon as prac-
ticable adopted also in England. And, in fact, it is already
used exclusively in some popular scientific class-books, and
a knowledge of it is required by Examiners in Physics and
Chemistry.
inconsistency, as the law now stands, whilst the restriction is removed
against contracting in terms of the Metric System, any person using
such weights and measures for the purpose of bujdng and selling in
shops and other places subject to the visits of Inspectors of "Weights
and Measures, or having them in his possession, is liable to have them
seized and to conviction and forfeiture.' Prof. Leonb Levi, 7%eofy
and Practice of the Metric Systenit p. 6.
M
158
APFENDIS in.
' l?. The Metric Syatem is bo called &om the French wOrd
meh'e (derived from the Greek Tnetron, ' meaanre '), the name
(pYcn to a line of a certain length (39'37 inches, rather more
tban a yard), which was fixod npon in 179y by the French
Legislature as the standard unit of linear measure, and
which was at that time Bapposed to be the
E ten-millionth part of the distance firom the
Equator to the Pole. It has been since
foond, however, that the measurement of the
Earth's circumference then made was not
qnitd correct. And, conseciaently, the Metre,
as originally determined by that meaanre-
ment, is really an arbitrary length, like the
English imperial yard.
18. The Metric System has fonr prindp&l
Dnits, all depending on the metre.
1. The Metro (39'37 incliea) is the unit of
meaanres of length.
2. The Are (120square yarde), the square
of ten metres, is the nnit of measures of
surface.
3. The Litre (61 cuJic inches), the cube of
the tenth of a metre, is the nnit of measures
of capacity.
4. The Gram (15J graiia) is the unit of
measures of weight, and is the weight in
vacuo of so much water at its greatest
density as would fill the cube of the hun-
dredth part of a metre.
19. The standard Metre is a platinum bar, and the
standard Kilogram (p. 161) a platinum cylinder, which are
preserved carefully in the H6tel des Archives at Paris.
Exact copies of them are deposited at tiie ConserTalioin
fc_.|
THE METRIC 8TSTEH.
159
des Arts et Metiers, and are nsed to verify tlie metric stan-
dards for foreign countries. But England possesses two
platinnm copies of the standard Metre, deposited with the
Boyal Society in London, and a platinum copy of the stan-
dard Kilogram, deposited at the Standard Department.
Besides these, brass copies of the Metre, Kilogram, and
Litre, have been carefully made, and presented by the
French to the British Government, and are now deposited
at the office of the Warden of the Standards.
20. Each unit has its decimal multiples and sub-
multiples, as follows : —
Length
Surface
Capacity
Weight
1000 kilometre
• • •
kilolitre
kilogram
100 hectometre
hectare
hectolitre
hectogram
10 dekametre
• • •
dekalitre
dekagram
1 MBTSE
ABB
I.ITBB
OBAM
*l(s^) decimetre
• • •
decilitre
decigram
•01 (=ife) centimetre
centiare
centilitre
centigram
•001 l=jks) millimetre
• . •
millilitre
milligram
21. The following are the tables of measures employed in
the Metric System, with their respective units.
1. Measures of Length or Linear Measure.
The unit of Linear Measure is the Metre=S9'd7 inches,
or 3-28 feet, or 1*09 yard (more correctly 39-3708 in.
=3-2809 /5.=l-0936 yds.).
10 millimetres make 1 centimetre.
10 centimetres
10 decimetres
10 metres
10 dekametres
10 hectometres
10 kilometres
})
a
ii
ti
It
»»
1 decimetre.
1 metre.
1 dekametre.
I hectometre.
1 kilometre.
1 myriometre.
Hence, in order to reduce from one denomination to
another, the French arithmetician merely throws the deci-
mal point one or more places to the right or left as tihs^ ^^s«&
u%
i
160 APPENDIX in.
may reqtdre. Thus 08765*4321 me/m=98765432-l milUm.
= 9*87654321 myriom, ; whereas under the English system,
in order to reduce 987654321 inches to leagues, we should
have to divide by 12, 3, 5^, 40, 8, 3, successively, a very
laborious process.
N.B. The deJcametre (10 m, or 100 e^m.=32*8 ft, or
10*9 yds,^ is used as a chain in surveying, and is divided into
50 links, each containing 2 decim.
The kilometre (1000 m.= 1093*6 yds.) is nearly 5 furlongs
(1100 yds.), so that 8 Mlom,:=6 miles nearly.
The myriometre (10 Mlomstres or 10,000 m.) =50 furlongs,
or 6^ miles nearly (more nearly =10936 yds, or 6 J- miles).
11. Measwrea of Surface or Square Measure,
The unit of Square Measure is the Are or square
dekametre, that is a square of which the side is a deka-
metre=10 metres, and which therefore contains (p. 26)
100 square metres=119*6 square yards.
100 centiares (square metres) make 1 are.
100 ares ( = 10,000 square metres) „ 1 hectare.
m. Measures of Solidity or Guhic Measure,
The unit of Cubic Measure is the Stere or cubic metre
=61027 cubic inches, or 35*3166 cubic feet, or 1*30802
cubic yard, nearly.
10 decisteres make 1 stere.
10 steres „ 1 dekastere.
N.B. These measures are chiefly used for wood and
carpentry,
IV. Measures of Weight,
The unit of Weight is the Gram., which is the weight in
vacuo of 1 cubic centimetre of distilled water at its greatest
density, viz. at the temperature of 4° of the centigrade
thermometer=15'43234 grains or 15J grains, nearly.
THE METRIC SYSTEM. 161
10 milligrams make 1 centigram.
10 centigrams „ 1 dedgranu
10 decigrams ,, 1 gram.
10 grams : „ 1 dekagram.
10 dekagrams ,, 1 hectogram.
10 hectograms ,, 1 kilogram.
10 kilograms ,, 1 myriogram.
N.B. The kilogram or kilo, as it is often called,
=15432*34 grains=2^ lbs. Av. (15,400 grains) nearly, is
the weight nsnallj employed on Continental railways ; and
the half-ldlo (=l-nr ^^' -^^0 ^ ^^ generally used as a
weight on the Continent.
The centner=50 kilos. =771,617 grains=110i lbs. Av.
(771,725 grains)=l cwt. (112 lbs.) nearly.
The qiiintal=10 myriogr. or 100 kilos. =220^ lbs. Av.
=2 cwt. nearly.
The millier or tonne=10 qnintals or 1,000 kilos=2205
lbs., or 20 cwt., or 1 ton, nearly.
V. Measures of Owpadty.
The nnit of Capacity is the Litre or cuUc decimetre
=61*027 cnbic inches=l*76 pint.
10 centilitres make 1 decilitre.
10 decilitres „ 1 litre.
10 litres „ 1 dekalitre.
10 dekalitres ,, 1 hectolitre.
10 hectolitres „ 1 kilolitre.
N.B. The hectoHtre=100 Htres=176 pints=;=22 gallons,
or 2| bushels, nearly.
22. Since 1 decimetre=10 centimetres, therefore (p. 28)
a cubic decimetre or litre =1000 cubic centimetres. Hence
the weight- in vacuo of a litre of distilled water at its
greatest density is the weight of 1000 cubic centimetres
of such water, or 1000 grams, that is to say, the weight
of a litre of such water is 1 kilogram.
162 APPENDIX in.
In like maimer, since 1 metre=10 decimetres, therefore
the weight of a cnbic metre of such water is that of 1000
cubic decimetres, viz, 1000 kilos or 1 millier. Thus a
mass of rock 4 metres long, 3 metres wide, and 2 metres
deep, would contain (4x3x2=) 24 cubic metres, and fill
24 kilolitres ; and as this quantity of water would weigh
24 milliers, the weight of the mass in question would be
found at once by multiplying this weight by the number
which expresses the specific gravity of the rock compared
with water,
23. A metric quantity may be read in various ways, in
terms of one denomination or of more than one, at pleasure.
Thus 35*703 metres may be read as 35 metres 7 dedm,
3 millim,, or as 3*5703 deJcami,, or as 357 dedm, 3 millwi,^
or as -035703 hilom.
But, in writing a metric quantity from dictation, it is
necessary sometimes to insert cyphers, as in the following
examples : —
Thirteen kilometres, seven grams=13'007 kilometres
or 13007 grams ;
Seven hectohtres three centilitres= 7*0003 hectolitres or
700-03 Htres ;
Seven hectares six ares five centiares= 706-05 ares or
7-0605 hectares.
But it should be noted carefully that in Square Mea-
sure such an expression as 5-7 sq, m, means — ^not 5 sq,
metres 7 sq, decim., but — 5-7 (=5-i^^) sq. metres =b sq, metres
70 sq. decim, (since 1 sq. metre=^100 sq. decim.). Similarly
in Cubic Measure 5*07 cub. m. means 5y5^ cub, metres
=5 cub. metres 70 cub, decvm. And conversely, since
1 sq. twe/re=100 sq. decim, and 1 cub. wefre=1000 cub,
decim.^ therefore 9 sq. metres 5 sq. decim.=9'05 sq, w.,
and, in like manner, 8 cub, mMres 91 cvJb. (^ecm. =8-091
THE METBIO STSTEK. 163
24. Since the inetre=l*09 yard or ly\y yard, nearly, and
the half-kilo=ly\j- lb. Av., nearly, it follows, that when
goods are sold by the metre or half-kilo, the prices should
be 10 per cent, higher than when they are sold by the yard
or pound respectively. In like manner since the centner
(50 kilos.) =110^ lbs,, which is less than a hundred-weight
(112 lbs.) by If lb,^=-^^ cwt, the prices of goods, when sold
by the centner or millier (20 centners), should be -^^ less
than when sold by the hundred-weight or ton (20 cwt)
respectively, which amounts to a reduction of 2f c?. in the £,
' 25. The metre, half-kilo, centner, and millier, might bo
called the metric yard, metric povmd, metric hundred-weight,
metric ton, respectively. And the following names, corre-
sponding to the names of English measures, are given by
Prof. Levi, Metric System, p. 64.
Metric league (half-myriometre) » s-i miles,
mile (kilometre) s 1094 yards,
furlong (double-hectometre) a 219 „
. chain (double-dekametre) s 21*9 „
pole (half-dekametre) a 5*5 ,,
fathom (double-metre) s 6*56 feet.
■^i cubit (half-metre) = 1*6 „
hand (decimetre) « 3*9 inches.
26. The following is a table of approximate equivalents
in the English and Metric Systems, where great accuracy
is not required (Prof. Gralbraith, as quoted by Prof. Levi,
Metric System, p. 49).
Length.
1 metre » 3 feet 3 inches 3 eighths.
64 metres := 70 yards.
Linear, Square, and Cubic Measure,
10 metres «= 11 yards.
16 sq. metres » 12 sq. yards.
10 cub, metres =s 13 cub. yaxds.
164 APPENDIX m,
Lamd Measv/re.
1 are b 4 perches.
10 ares » 1 rood.
1 hectare » 2^ acres.
Weight.
1 kilograms 2J lbs. Ay.
30 grains =17 drams Ay.
Liquid a/nd Dry Measure,
4^ litres » 1 gallon.
1 hectolitre = 22 gallons.
27. The following table gives a more accurate list of tb
equivalents of the principal metric measures in terms c
English measures, and vice versd,
Measttbes of Length.
MUUmetre
ss
•03937 inch.
Centimetre
»
•3937 „
Decimetre
B
3*937 inches.
or
•32809 foot.
Metre
a
39-37079 inches.
or
3-28089 feet.
or
1-09363 1/ard.
Dekametre
«
10-93633 1/ards.
or
1-98842 pole.
Inch
a
•0254 metre.
Foot
a
•30479 „
Yard
a
•91438 „
Pole
»
5 0291 metres.
Chain (4 p.)
rs
20-1164 „
FurUng (lOch.)
1 =
201-1644 „
MUe
a
1609-3149 „
or
1-6093 Mom,
Measures of Surface.
Square decimetre « 15*50059 square inch.
Square metre « 1 19603 square yard,
or 10*76429 square feet.
THE METRIC STSTEM.
165
Hectare
a
2-47114 acres.
Are
-
•02471 a(?re.
Square inch
a
6-45137 square centimetres.
Square foot
«
9-28997 square decimetres. •
Square yard
=
•8361 sq. metre {oentiare).
Square pole
=s
•2529 ar«.
Bood
s
10-11678 are*.
Acre
a
40-4671 „
OP
•40467 hektares.
Measures of Solidity.
Cvbie decimetre »
61-02705 cubic inches.
Cubic metre
a
35-31658 cubic feet.
or
1^30802 cubic yard.
Cubic inch
a
16*38618 cubic centimetres.
Cubic foot
=>
28-3153 cubic decimetres.
Cubic yard
=
•7646 cubic metre.
Measures of Weight.
Oram
s
•56438 dr.
OP
•03527 ounce Avoirdupois,
OP
15^43234 grains.
OP
•64301 dwt.
Hectogram
«
3*52739 ounces Avoirdupois,
OP
3-21507 ounces Troy.
Kilogram
=
35-2739 ounces Avoirdupois.
OP
2^2046 pounds Avoirdupois,
OP
2-6792 pounds Troy.
or
•01968 hundred'Weighi,
MiUier
=»
•98420 ion.
Grain
^
•0648 gram.
Pennyweight
=
1-55517 „
Ounce Troy
=
31 '1036 grams.
Pound Troy
=
373-24196 „
Dram
. »
1*77184 gram.
Ounce Avoirdupois =
28-34954 grarns.
Pound Avoirdupois «
453-59265 „
Stone (14 lbs.)
=
6-3503 kilograms.
Quarter (28 lbs.
) -
12-70059 „
166
APPENDIX in.
Hundred-weiffht
Ton
B 50-80238 kilograms.
« 1016-0475 „
or 10-160476 quintals,
or 1-0160475 mUlier.
Measures of Capacity.
Centilitre »
•07043 ffiU.
Pint «
•56755 ?«^;^
Decilitre =
'17607 pint.
Quart =
1-13510 „
Litre =
1-7607 „
Gallon =
4-64041 litres.
or
•88038 quart.
Bushel —
S-6Z2SZdekalitr€S,
or
•22009 gal
or
36 3233 lUres,
Dekalitre =»
2-20096 „
Quarter =
2-90586 hectolitres.
Hectolitre =
220096 „
or
290686 dekalitres.
or
2-761208^^5.
or
290-6S6 lUres,
or
•343901 gr.
167
NOTES AND EXAMINATION-PAPEES
OW
AKITHMETia
NOTES.
Note I.
Casting out the Nines, as a method of Proof for Multiplication,
depends on the two following considerations: —
(L) Anj n<* divided bj 9 leaves the same remainder that would be
left if the sum of its digits were divided by 9.
Thus, 687-5-9 leaves 3; and (6 + 8 + 7) -r 9 leaves 3.
(ii.) If each of two no* be divided by any n<», say 9, and the product of
their remainders be taken, this product divided by 9 will leave the same
remainder that would be left if the product of the two no' were divided
by 9.
Thus, 1547-4-9 leaves 8, and 687—9 leaves 3; then, (8 x 3)-i-9 leaves
6, and (1547 x 687)— 9 also leaves 6.
The first of these considerations will appear just from the following
illustration.
10, or 100, or 1000, or any other power of 10, is an exact n'* of nines
+ 1; therefore*
80 is an exact n" of nines + S,
600 is ditto +6,
680 18 ditto +6 + 8,
687 is ditto +6 + 8 + 7;
80 that 687 -^9 leaves the same remainder as (6 + 8 + 7)-^9.
It is evident, then, that to ascertain what remainder would be left
after dividing any n^ by 9, we need only sum the digits of the n% and
cast out 9 as often as it arises in the addition.
The second consideration may be illusttaX^^ "V^^ \k^ i^^s^ro^^
example:—
168 NOTES.
Since 1647 = 171 nines + 8,
and 687 « 76 nines + 3,
therefore, 1547 x 687 is equal to
(171 nines + 8) x 76 nines, [which gives an exact n* of nines'}
+ (171 nines + 8) x 3 ; [which gives an exact n* of nines + 8x3];
evidently, therefore, the whole product is an exact n*> of nines + 8 x 3, oi
+ 24, or + 6; the 6 being obtained by adding the digits 2 and 4.
Note II.
When a divisor is composed of two or more factors, and the quotient
b found by using those factors successively, the remainders after the
several divisions may be converted into the full remainder in the manner
employed in the following example:^
Divide 39711 by 35, or by 5 x 7.
5)39711 Or, 7 )39711
7) 7942 ...1-I 5x4+1 5 ) 5673 ...0-| 7 x 3 +
1134.. .4 J =21 rem. 1134...3 J =21 rem.
Quotient, 1134|i,or 1134f.
Dividing by 5 first, the successive remainders are 1 and 4; or, divid-
ing by 7 first, they are and 3; and to find the entire remainder, we
multiply the first divisor by the second remainder, and to the product
add the first remainder.
The reason of this procedure may be shown thus:—
We are required to find how many thirty-fives are contained in 39711
units. Dividing first by 5 units we find that 39711 is =7942 fives -^
1 unit; and then dividing the fives by 7 we find them =1134 thirty-fives
+ 4 fives; so that 37911 units are equal to
1 134 thirty-fives + 4 fives + 1 unit, = 1 134 thirty-fives + 21 units ;
= 1 134 thirty-fives + §} of 35 ; = 1 134|| thirty-fives.
In the second form of the division we have as the first remainder:
in such instances, the second remainder placed over the second diyisor
gives the fractional part of the quotient in a simpler form.
Note IIL
Strictly, in reducing £37 to shillings, we multiply— not £37 by 20,
which would produce £740, but 37 by 20; the reasoning is that £37
contains 20 times as many shillings as pounds.
Note IV.
The multiplication of dimensions is frequently performed by what is
called the method of Duodecimals, which subdivides both square feet
^iod cubic feet into denominations called primeS) secofi^As^tKirck^&c.; IS
NOTES. 169
superficial primes being = a square foot, 12 cubic primes = a cubic foot,
and, in both cases, 12 seconds :=a prime, 12 thirds = a second, &c
Primes, seconds, &c., are marked thus,
15 sq. ft. r 10" 6'"; 15 cub. ft. 7' 10" 5'".
In the first of these expressions the seconds evidently are square
inches, for they are l^iihs of a square foot; and if to these we add the
7 primes, or twelfths of a sq. foot, « 84 one-Kundred-and-forty-fourths of
a sq. foot, we have 94 sq. inches, and the whole expression is equivalent
to 15 sq. ft. 94^ sq. in.
In the second of the expressions the thirds are evidently cubic inches,
for they are l72Sths of a cubic foot, and if to these we add the 7 primes
and 10 seconds, which are = 1 + i5. = l^ + i?2. of a cubic foot, we
12 144 1728 1728
have 1128 + 5 cubic inches, and the whole » 15 cub. ft. 1133 cub. in.
Suppose, now, it is required to find by duodecimal multiplication the
area of a rectangular surface, 37 ft. 7 in. by 5 ft. 9 in.
Here, since 37 ft. 7 in. = 37^ ft., if the rect-
angle were 1 ft. broad the area would be 37^
sq. ft., or 37 sq. ft. 7'; then, as the breadth is
6^ ft., we multiply 37 sq. ft. 7' by 5 units 9
twelfths, as follows: — Placing the greater di-
216 so ft T 3^' mension over the less, we first multiply 37 sq.
ft. 7' by 5, then we multiply the same quantity
by 9 considered as twelfths, and by setting the remainder, arising from
a twelfth of 9 times 7, one place to the right of inches, and carrying 5 to
the next product, of which in like manner we take a twelfth, we shall
evidently have 9 twelfths of 37 sq. ft. 7' =-28 sq. ft. 2 twelfths of a sq. ft.
3 twelfths of a twelfth of a sq. ft.
The entire product is 216sq. ft I prime 3 seconds.
If I2ths of an inch, commonly called parts, occur in either of the
factors, the duodecimal multiplication is performed in the same way.
Let it be required to multiply 28 ft. 9 in. 6 pts. by 1 1 in. 9 pts.
It should be observed that the
annexed process, which is con^
ducted in the same way as the
preceding one, is equivalent to
^Ssq. ft. 2' 3" 7'" 6"" finding first — of the multiplicand,
then — or -— of it, and that we do not really multiply ot\a ^qt^jw^J^
144 16
quantity bj another, which would be absurd.
ft.
pr.
37
7
5
9
18^7
11
28
2
3
ft.
pr.
sec.
28
9
6
11
9
6
26
4
8
1
9
7
1
6
170 NOTES.
Note V.
For a demonstrative arithmetical example of the process of finding
the greatest common measure of two numbers, see Hunter's Art of
Teaching Aritlimetic, p. 64. A very slight acquaintance with Algebra
will enable the student to understand the following illustration of the
general Rule for finding the o.cm.
Let it be required to determine the o.c.m. of 1275 and 561.
The o.CK. of 1275 and 561 evidentlj cannot exceed 561, and must be
s=:561-^some factor of 561. Let x denote that factor. Therefore, the
561
O.C.M. of the proposed no* will be — , when x has the least yalue that
X
kft\
allows --- to measure 1275.
X
We have to find, then, the least valae of x making 1275 -r—-, or
X
i£Z^ a whole n'.
561
Now, iH5f-2* + i|?of *; sothat — of arisawholen*.
• 561 651 561
/, — of A=a whole n*, which we may call b ;
153 J '
:, A-H? of B^B^+ii of B5
102 102
Similarly, ^ of B = ; /. b = i^ of c, = 2 c exactly.
Now, we should get B = a whole n*, whatever whole n« we might choose
for the value of c ; but we must take c^^ 1, the lowest whole n", that we
may obtain the lowest integral value of x.
Hence, !275£^1275 ^^ 661 ^^ 163 ^^ 102 ^^ ^ ^ 127_5
561 561 153 102 51 51
.'. -^ = — , or, — = 51, the g.cm. required.
DdI di X
From the above analysis, then, it appears that the o.cm. of two no*
is obtained by dividing the greater by the less, then the less by the
remainder, and so on as prescribed by the Rule.
To determine the g.cm. of three no*, find that of two of them, and
then that of the result and the third number. Thus, the o.cm. of
12528, 16182, and 13804^ will be founds 58 *, Ux that of the first two
DO' 18 522, and that of 622 and 13804 is 58.
To £nd the o.cm, of fractional qnant\l\«B, tA^icft^sKccL^lft^ of 9% and
K0TE8. 171
i9|, express them as fractions haying a common denominator, then find
the o.CM. of the numerators, and under it write the common denomiua*
tor. The result for the supposed example will be -^, which is con-
tained 15 times in the first n<* and 34 times in the second.
Note VL
For the conversion of a mixed circulating decimal to a vulgar frac-
tion, the following rule is self-demonstrating: — Multiply the given
decimal by 10, or 100, or 1000, &c, according as there are one, two,
three, &c., decimal places before the circulating period; express the
result as a mixed fraction, and then divide it by the 10, or 100, &c., pre-
viously used as a multiplier, which will evidently restore the value of the
given expression.
Thus, to convert *034 and '27345 to vulgar fractions : —
(L) •034xl00=3-4=3|;
and 3|x-i-=-?l.
* 100 900
(iL) '27346 X 1000=273-45 -273||=273^;
and 273^x-L-=522i=i!i.
^^ 1000 11000 1375
What is further included in the usual Rule has reference to an easy
method of multiplying by the denominators 9, 99, 999, &c.
and 273 x 99 being=273 x (100-1),
27300-273 + 45 27345-273
we have
99000 990U0
27072 3008 376
9U000 11000 1375
NOTB VIL
The series proposed for calculation in Ex. 4k7, 48, is one by which
the ratio of the circumference of a circle to its diameter may be
approximately computed. See Colenso's Plane Trigonometry, Part IL
p. 7. The result signifies that the circumference of any circle is nearly
3*14159 times the diameter.
The series proposed for calculation in Ex. 4k7| 50, is that whereby
what is called the base of the Napierian system of Logarithms is
approximately computed. See Colenso's Plane TrigouoTOfctnj^ 'SwN.^
p, 121, or Hunter's Treatise on LogcwithmB^ p. 55. T\ia xwdXx «v^>S«a
172 NOTES.
that the Napierian Logarithm of any given number is that power of
2*7182818 which when calculated produces the given number.
Note VIIL
Questions in Proportion can always be worked independently of the
artificial Rule of stating, and though sometimes not so conveniently, yet
always in a more satisfactory way as regards simplicity of demonstra-
tion. It will appear from the following examples that a knowledge of
the first principles or fundamental rules of Arithmetic is sufficient for
the solution of all problems in the Rule of Three.
(1) If 15 lbs. of salt cost Is. Qd,, what cost 25 lbs.?
Cost of 15 lbs. = 18c?.
„ 1 lb. =^ of \Sd.
„ 25lbs.«f|of 18rf.
18(/.x25^g^ ^ g^2,. 6d. Ans.
15
(2) If 25 lbs. of salt cost 2s. 6(/., what quantity cost Is. 6d,?
No. of lbs. for S0c?.«25 lbs.
„ l(/.=^of 25lb8,
„ 18c?.«J§of 25lbs.
?il^?i^«5 lbs. X 3 = 15 lbs. Ans.
SO
(3) What is the coach fare for 130 miles at the rate of £l 9s. id.
for 85 miles?
Fare for 85 miles = 29|».
„ 1 mile =r^ of 29|«.
„ 130 miles = ^fto of 29|«.
88.. X 130^88.. X 26^^^ j^^ ^^^^
3x85 51
(4) If 112 sheep were grazed in a field for 9 days, how long might
84 sheep have been grazed in the same field?
Time that 112 sh. were grazed b9 da.
„ 1 sh. might be grazed = 112 times 9 da.
„ 84 sh. „ „ s=^ of 112 times 9 da.
9 da. X 112 _ , . -„ , .
~3da. X 4 = 12 da. Ans.
84
(5) A person completed a journey in 32 days, travelling 8 hours a
day ; how long would he have taken to do the same, travelling only
6 hours a day?
No. of days at 8 hrs. a day = 32 da.
„ at 1 hr. a day =8 times 32 da.
„ at 6 hrs. a day=^ of 8 times 32 da.
6 a *
NOTES. 173
(6) Three partners with a joint stock of £lOS6 Us, Gd. gain
£287 6s,; what share of the gain falls to one of the partners whose
stock is £365 I7s.?
Gain on £1036 11*. 6d, (or 41463 sixp.) = 6746«.
„ on 1 sixp.= of 6746*.
^ 41463
„ on £365 17*. Od. (or 14634 sixp.)
14634
41463
6746.9. y 14634 574G*. x 1626
41463 4607
of 5746*.
=^£101 8*. Ans.
(7) If 10| lbs. of sugar cost 4\^s.f what will 3| cwt cost?
Cost of lOf lbs.=4}i*.
„ of 1 lb. =^1 of l^s. « JL of 7«.
" 75 16 16
„ of 112x31 lbs. =:llii^3 of 7«.
" * 16
7*.xll2xll^49...vll^^Q 19*. Sd. Ans,
16x3 3
Note IX.
In calculating the amount of any sum of money, by compound in-
terest, for any n" of years, at 4 per cent per annum, we add to the
original principal — of itself to obtain the 2nd principal, then to this
principal we add - — of itself to obtain the 3rd principal, and so on.
Now, adding to any n« — - of itself is the same as multiplying it by
Iioo» O' ^7 ^'^^f and accordingly, the amount of £750 for 3 years, at
4 per cent per annum., comp. int. might be found thus:—
£750 X 1*04 x 1-04 x 1'04,
=£750 X 104»=-.£750 x 1124864,
«= £843-648.
Similarly, the amount of £750 for 4 yrs. at 5 per cent would be
£760 X 1*05*. And, generally, to find the amount of £p, by comp.
interest, for any n* of years, at any annual rate, we may first add a
hundredth of the rate to 1, then raise the sum to that power which is
denoted by the n^ of years, and then multiply by p.
Suppose that, in this way, we have to find the compound interest of
£95 6*. 8d, for 3 yrs., at 5 per cent per ann., payable half-^^^xV^x —
the rate is here intended to denote 2| per cent, i^t YksS^-yeAXt l^t ^\is^-
jean.
174 NOTES.
We have accordingly to find the 6th power of 1-025; and' this we
could obtain at once from compound interest Tables ; or we could very
easily calculate it from a Table of Logarithms. The simplest form of
the arithmetical process is as follows ; the divisor 40 determining the
interest in each case, because 2^ is ^^ of 100.
40)1 025 Amt. of £l for 1 hf. yp.
'025625
40)1050625 Do. „ 2 do.
•0262656
40)1-0768906 Do. ^ 3 do.
•0269223
40)1^1038129 Do. ,, 4 do.
•0275953
40)1-1314082 Do. „ 5 do. *
•0282852
11596934 Do. „ 6 do.
Hence the compound interest of £l, at the end of the 3rd year, is
£-1596934 ; which multiplied by 95| gives the comp. int of £95 Ss, Sd,
= £15-2241, or £15 45. 5-7Sd, Ans.
Now, suppose it is required to find what principal at 2^ per cent, per
annum, comp. int., will in 6 yrs. amount to £110 3^. 5d. : that is, what
IS the present worth, by comp. int., of £110 3«. 5(2. payable in Gyra : —
we have
l-025«xp= 110170833;
/. 110-170833-rl-1596934 = £95. Ans.
Again ; let it be required to find at what rate of comp. int. £95 will
amount to £110 8s. dd. in 6 yrs. : —
110-170833-5-95 = 1-1596934, the 6th root of which may be found
by logarithms = 1-025; or, ^/l•1596934 = l•0768906, the cube root of
which is 1*025. Hence the rate is 2^ per cent. Ans,
Lastly ; to find in what time £95 will amount to £110-1 70833, at 2^
per cent, per ann., comp. int. : — Here we should ascertain by logarithms
what power of 1*025 is equal to 1-1596934 ; but when the time is an
exact n<> of years, as in this instance, it would be found by raising
1*025 through consecutive powers till the required amount of £1 is
found equal to the 6th power, denoting the time to be 6 yrs.
NOTB X,
A Kule called Equation of Payments is introduced in some treatises
on Arithmetic. It teaches how to ascertain the single time at which
two or more debts, due at different times, might be discharged by one
pajment of the Bum of the debts. It is merely a particular application of
NOTES, 175
the principle of Discount; and it is given in two forms, according to
true discount and mercantile discount, respectively.
JSxamp. I owe £1085; of which £651 is due 5 months hence, and
£434 is due 8 months hence; how many months hence would one pay*
ment of £1085 discharge hoth dehts, reckoning the use of money
worth 5 per cent per annum?
We compare the several sums by means of their present values, con-
sidering that the discount on £651 for 5 months added to the discount
on £434 for 8 months, should bo equal to the discount on £1085 for the
time sought.
Now, according to Mercantile Discount, we have
5
1200
of £3255= int. of £651 for 5 months;
and — of £3472 =int of £434 for 8 months;
1200
^ of £6727 =int. of £1085 for 6*2 months. Ans.
1200
because 6727 -f- 1085 « 6-2.
This method is evidently independent of the rate of interest; and
hence, for equating terms of payment according to mercantile discount,
we have the following
Ordinary Rule. Multiply the several debts by their times in any uni-
form denomination, and divide the sum of the products by the sum of
the debts.
Thus, the above process is reduced to the following: —
651 X 5 = 3255
434 X 8 = 3472
1085 )6727^
6^ months. Ans,
Tho meaning of which is, that as the int. of £651 for 5 months is
that of £3255 for a month, and the int. of £434 for 8 months is that of
£3472 for a month, so the int. of £6727 for a month is that of £1085 for
6} months.
Bat secondly, according to True Discount, we have
1- of £5, or £2^= disc, on £102^ for 5 mths.
12 » 12
or -^=disc. on 1;
49
1. of £5, or £3|=disc. on £103| for 8 mths.
0' r-=disc. on li
31
176 NOTES.
/. ^ ^£IS% is the disc, on £651 for 5 mths.
— ■*= 14 is the disc, on 434 for 8 mths.
31
£27f is the disc, on £1085 for the time sought.
We have to find, therefore, in what time £l057f would produce
£27f interest, or £7404 would produce £191.
7404 : lOO-i ,^ . ^,,g .
5 . J9J [ :: 12 mo. : 6|i| mo. Ana.
Tins answer, equal to ahout 6*19 months, is a little less than 6.2, the
answer found according to mercantile discount; hut as the method of
true discount is much more laborious than the other, and in most prac-
tical questions gives a result very little less than the other, it is generally
sufficient, as it is more convenient, to follow the ordinary rule.
The Rule for equating according to true discount nay be given as
follows: —
Find for each of the debts the discount that woul& reduce it to its
true present value; then find the time for which the sum of the dis-
counts would be the true discount on the sum of the debts.
For a discussion of the principle of Equation of Payments, see
Hunter's Art of Teaching Arithmetic, p. 79.
Note XI.
In Paper IX. will be found a variety of Questions relating to the com-
parison of the money of different countries. This subject is fireqnetitly
treated in books on Arithmetic under a special Rule called Exchange,
The Par of Exchange is the intrinsic value of the coin of one country
as compared with a fixed sum of the money of another. The Course of
Exchange is the variable sum of the money of one country actually
given for a fixed sum of the money of another.
Thus, France exchanges with England a variable number of francs,
averaging about 25*30, for the pound sterling; for the actual Course of
Exchange, being dependent on the course of trade, is in almost con-
tinual fluctuation. Moreover, as in England gold is the adopted
standard of value, and France has a silver standard; — as also the values
of gold and silver are not always in the same proportion, and each
mctul has not always the same value in both countries, — the Par itself is
not invariable.
Arbitration of Exchange is the estimation of the rate of Exchange
implied in the purchase of indirect Bills of Exchange, Bullion, Coins,
&c., in one country, as compared with their sale in another.
Thus, to Bnd what arbitrated rale o^ 'E&^\iaxL^^ \s^ ^^bUshed between
WTJUB. 177
London and Paris bv bills on Vienna bought in London at 10 florins
1 krcutzer per £ sterling, and sold in Paris at 254 francs per 100 florins;
a florin being = 60 kreutzers: —
Here we have given £1 = 601 kreutzers, and 600 kreutzers =25.4
francs; /. 1 kr. = ^ fr.,
and 601 kr. =25'4 fr. x IgJ^s 25*44 fr. per £, Ans.
Again; to find what arbitrated rate is established between London
and Paris by the purchase of gold in London at 77*. lOjrf. per ounce
standard, and the sale of it in Paris at 4 per mille premium: an ounce
Troy being =31* I grammes, and 1000 grammes of English standard
gold bemg worth 3151 francs : —
Here wo have 311 grammes =10 oz., or 1 gramme= — oz.|
311
/. 1000 grammes = 1522? oz.,
1000 grammes bought in London for 77j*. x — — ;
o 1 L
1000 grammes sold in Paris for 3151 frs. x 1'004;
6230000^ : 20*. :: 3163*6 frs. : 25*27 frs. nearU Ans.
ft X 3.11
178
EXAMINATION-PAPERS.
Paper L
Questions on the Introductory Pages,
1. (a) Explain the principle bj which the decimal system of nota-
tion is made capable of expressing any number whatever.
(6) Distinguish between the arts of Notation and Numeration.
2. Add Thirteen thousand thirteen hundred and thirteen to Seventeen
thousand seventeen hundred and seventeen,
3. Subtraction may be performed (a) for the purpose of diminishing
a quantity by taking away some quantity it contains, or (b) for the pur-
pose of comparing two quantities as to their absolute magnitudes.
Give properly distinctive names for the results in these two cases.
4. (a) If two numbers be equally increased, how is their difference
affected ? A fathei is 3 score and 5 years old, and his son is 37 ; what
is the difference of their ages? and what will be the difference of their
ages 10 years hence? (b) Apply these considerations to explain the
process of borrowing ten and carrying one in subtraction.
5. (a) What name is given to two or more numbers connected by
multiplication? (6) Show how six sevens are equal in amount to
7 sixes, (c) Show why multiplying successively by 6 and 7 gives the
same result as multiplying by 42.
6. What are the methods commonly used for proving the accuracy
of multiplication? How might division (if the pupil understood that
p:ocess) he used as a trial of correctness in multiplication?
7. (a) Divide 27564 by 21 in two ways: — resolving 21, first, into
successive divisors 7 and 3, and secondly, into successive divisors 3 and 7.
(b) Explain by reference to your work the usual process of finding the
full remainder by means of the two partial remainders.
Paper IL
Questions on Articles 1 to 20.
/• In reducing £7 to shillings vrhat multiplier, strictly considered, do
we employ? Explain.
BXAMINATION-PAPEBS. 179
8. In dividing a concrete quantity by an. abstract number, as for
example in finding the 8th part of £3 7^. &d. (Colenso, p. 24), which of
the expressions is properly the quotient? and why?
3. How would you reduce crowns to guineas? florins to crowns?
sovereigns to guineas? yards to English elis? lbs. Avoirdupois to lbs.
Troy?
4. (a) Under what conditions may one concrete quantity be added
to another? subtracted firom another? divided by another?
(6) Why cannot one concrete quantity be multiplied by another? -
5. (a) How is the square measure of a rectangular surface found
from its length and breadth? If the length be 5 feet, and breadth 4 feet,
is the area = 5 ft. x 4 ft. ? Explain.
(b) How is the width of a rectangular space found when the length
and area are given?
6. (a) How is the cubic measure of a rectangular solid found from
its length, breadth, and height? Suppose the dimensions are 8, 6» and
2 feet : — explain the process of finding the solidity.
(b) How is the height or the thickness of a rectangular solid found*
when its cubic content and its length and breadth are given p
Paper III.
Questions for Illustration of Ex, 17*
1. (a) A man's yearly income is known:— How would you find the
sum he must spend weekly, so as to lay by a given sum' at the year's
end?
(b) Given, a man's daily income and his yearly expenditure: — How
do we find his weekly saving?
2. The sum of 3 crowns, 3 florins, and 3 pence, is equal to 3 times the
ram of a crown, a florin, and a penny, that is, 3 times S5d, — Apply this
tfonsideration to the solution of Exs. 60, 61, and 62, in Set 17.
3. If £342 is to be multiplied by 242, and the product divided Ly
11, 8, and 4, successively, the effect of the whole may bo sjrmbolically
, ^, £342x242 i • v, i, w ^ £171x22
expressed thus, — — ,- which, by cancellmg, becomes -—
and by further cancelling becomes ^^ 71 x 11 ^£1881^^235 25. Gd.
—Apply this mode of treatment to the solution of Exs. 45, 55, and G7>
in Set 17.
4. How do you find the average value per yard of a quantity of
gocds, consisting of 20 yards at 125. 6d. and 35 yards at 9«. IQd^l —
Would the result be affected by the alteration oi taSdu^oi^-^xJcL eft. %w2s\.
180 EXAMINATION-PAPEBS.
of the given quantities, making them together « 11 yards? — Solve Exs.
42 and 51, in Set 17.
6 In Ex. 63, Set 17, show that the result equals tl — — years, or
_, or -, of a ycar,=2ij years; and explain the following process:—
24 12
12)365 da. 6 hrs.
^0 lOj
Ans. 2 yrs. 334 da. 19| hrs.
6> (a) Kcduce 4 men 7 boys to an equivalent number of boys, sup-
posing a man equivalent tO 3 boys.
{b) lieduce 7 men 12 women 5 children to an equivalent number of
children, supposing 2 women equivalent to a man, and 3 children equi-
valent to a woman.
(c) Apply the above species of reduction to the solution of Exs. 58
and 65, in Set 17.
7. (a) If the number 365 is to be divided into four parts, three of
them equal, and the fourth 95 less than each of the others; how many
times the first part would make 365 + 95?
Apply a similar mode of inquiry in the solution of Ex. 64, Set 17.
(6) If the sum of two numbers is 135 and their difference is 95, show
how each number may be found.
Divide a sovereign between Harry and George, giving George 20d.
less than Harry.
In a certain manufactory 7 men and 5 lads are employed, and each
oi the men earns weekly 17^. 6d, more than each of the lads. Now,
if 2 of the men be absent for a week, the total amount required to
pay the ten persons for that week will be £S 28, 6d, How much was
earned by each man per week? — Ans, 258,
Paper IV.
Questions on Chapters IT, III, and IV,
1. What is meant by a common measure of two or more numbers?
How is their g.c.m. ascertained?
2. What is meant by a multiple of a number? How do you find
the L.C.M. of two or more numbers?
3. Show that the product of two numbers divided by their o.cic
gives their L.C.M.
4. Find that the g.cm. of 11310, 12354, and 64090, is 58.
5. How do you find the g.c.m. of numbers all or partly fractional?
Fwd the O.C.M. of 26 J, 28 J, and 29|=^.
JEXAMII7ATI0N-PAPERS. 181
6. How do you find the l.c.h. of numbers all or partly fractional ?
Find the L.C.M. of 10|, 6|, and 4^=4042|.
7. What is a fraction? Is 3 farthings an integral or a fractional
quantity ? Define a concrete fraction,
8. What arithmetical operation is signified by the line separating
the terms of a fraction? What is an improper fraction, and how is it
reduced to a proper form?
9. What rule of fractions is anticipated in reducing a mixed frac-
tion to an improper one?
10. Why is it necessary that fractions should be of one common
denominator for addition or subtraction?
1 1. (a) Show that multiplying the numerator of a fraction is equi-
valent to dividing the denominator, and that dividing the numerator is
equivalent to multiplying the denominator.
(b) Hence show that the value of a fraction is not changed by mul-
tiplying or dividing both its terms by any one number.
12. What name is given to a fractional expression of the form f of |?
Which quantity is thus denoted to be a multiplier of the other?
13. (a) Prove the rules for multiplication and division of fractions:
exemplify with § and |.
(^) What does multiplication by a fraction strictly mean?
14. Explain the meaning of such a fraction as — ' — -
•t^ L 4 St
15. (a) A certain quantity, A, is given:— If it be | of another quan-
tity B, how would you find B? If it be half as much again as J?, how
would you find B?
{b) A number increased by its 5th part amounts to 30: how would
you find the number?
(c) A number diminished by its 5th part becomes 24: how would
you find the number?
16. Distiuguish between decimal and vulgar fractions. What is the
special utility of decimal fractions ?
17. (a) State and prove the rule for pointing in multiplication of
decimals, (b) How do you determine the local values of the quotient
figures in division of decimals ?
18. (a) What are circulating decimals? (b) Distinguish those
vulgar fractions that are convertible into termmating decimals ; and
shew that all the others are convertible into recurring decimals.
182 EXAMINATION- PAPERS.
Paper V.
Supplementary Questions in Reduction of Measures*
1. Reduce 22870062 square inches to acres, &c.
f 12) 22870062
ll2) 1905838... 6) ^^ .
\ } 196 in.
9 ) 158819 .. .10 3
80^) 17646... 5 ft
121 r 11 )70584 qr. yds.
111)^6416 .. 8-|^4i qr. yds. = 10 yds. 2ft 36 in.
4 0)583. .. 3j 5 126
4) 14.. .23 po.
Arts. 3 ac. 2 ro. 23 po. 10 yds. 8 ft 18 in.
2. Reduce the preceding result to square inches.
3 ac. 2 ro. 23 po. 10 yds. 8 ft 18 in.
4
14 ro.
40
583 po.
30J
145J
17500
17645J yds.
9
158819} ft
12
1905837
12
22870062 in. Ans,
^ 3. Reduce 1254492 sq. in. to sq. poles, &c.
4. Reduce 1 ac. 3 ro. 39 po. 14 yd. 5 ft. to sq. inches.
5. Reduce 123456789 sq. inches to acres, &c.
6. Reduce 2 ac. 3ro. 13 po. 14 yd. 5 ft 100 in. to sq. inches.
7. Reduce 9532482 sq. inches to acres, &c.
8. Reduce 2 ro. 22 po. 14j yd. to sq. feet
9. Express 22 gq. po. 2 yd. 4 ft. 72 in. in the denomination of sq.
yards.
10. An imperial gallon measures 277*274 cubic inches; how many
gallons would a vessel contain of which the capacity is 196 J cub. feet?
11. The length of a wall, according to the French metrical system,
Js 9 metres 4 decimetres 8 centimetres; reduce this to English feet, the
IcDgtb of the metre being 39 '3? I mchea.
EXAJlINATIOX-rAPERS.
183
12. Reduce 13 feet to metres.
13. How many decametres correspond to 1760 yards?
14. A chain 66 feet long is divided into 100 equal parts called links.
Kedace an acre to square links.
15. A rod of brickwork, viz. a square pole, or 272 J square feet, has
a standard thickness of a brick and a half: — If apiece of brickwork be
48 feet long and 22 feet high, and 2| bricks thick, to how many rods of
standard thickness is it equivalent?
Paper VL
Questions on Bath, (See Art 73.)
1. If the ratio of Z to 3f is 5 : 8, and that of Jlf to iV is 6 : 7 ) what
is the simplest form of the ratio of L to N?
Here Z is 1 of M, and M is -o( N:
8 7 '
/. Z is 5 of 5 of iV=l-^ of N, Ans.
8 7 28
Or, Z is to A'' as 15 : 28. -4ms.
2. M buys 15 cows and 130 sheep for a certain sum, and N buys
9 cows and 175 sheep, at the same rates as M, for the same sum. Com-
pare the values of a sheep and a cow.
Since N has 6 cows fewer than M,
but has 45 sheep more than M,
and both persons pay the same amount,
it is evident that 6 cows are worth 45 sheep,
6 2
or 1 sheep worth — , or — , of a cow,
45 15
or the values of a sheep and a cow are as 2 r 15. Ans.
3. One vessel contains a mixture of 16 pints of brandy and t? of
water; another contains 24 pints of brandy with 11 of water. Co:r
pare the strengths of the two mixtures.
Ist mixture 21 pints, 16 of which are brandy,
2nd „ 35 „ 24 „ „
/, the strengths are — and — ,
^ 21 35
2 3
or as - to ■-, or as 10 : 9. Ans,
3 5
184 EXAMINATION-PAFERS.
^ 4. A boat whose speed was 9| miles an hour sailed from A toB,
a distance of 65 miles; and a second boat, which left A 2\ hours after
the first, arrived at i? 5 minutes before the first. Compare the rates of
sailing.
5. A and B buj oranges at 10 for a shilling; A retails them at 9 for
a shilling, and B at lid. for a dozen. Compare their gains on selling the
same number of oranges.
A
6. If ^*« rate of profit is -of 5'*, and for every guinea that ^ gains
C gains a sovereign, compare the profits of A and C
7. A sum of money is so divided among Roger, Henry, William,
and Thomas, that R. gets 3c/. as often as H. gets 2|</., H. gets Zd, as
often as W. gets 4|rf., and W. gets Ad, as often as T. gets 3|</. Find the
direct proportion of the four shares.
8. If 3 men and 11 boys, working together, can do 5 times as much
work per hour as a man and a boy together, compare the work of a boy
with that of a man.
9. One vessel M contains a mixture of 27 gallons of wine and 11 of
spirits; another vessel A^ contains a mixture of 43 gallons of wine and
14 of spirits. Compare the strengths of the two mixtures, supposing
the strength of spirits to be three times that of wine.
Paper VIL
Questions on Averages,
1. In a school register of daily attendance the numbers for a
certain week were — Monday 83, Tuesday 80, Wednesday 75, Thurs-
day 80, Friday 77, Saturday 72. What was the average daily attend-
ance?
2. A tradesman's receipts of money in one week were — Mon.
33/10|, Tues. 26/6, Wednes. nothing, Thurs. 10/81, Fri. 43/111. Satur-
day 30/10. What was the average daily receipt?
3. The quantities of maize raised in the United States, In three suc-
cessive yearj, were— 494618200, 421953000, and 417899000 bushels.
What, in British currency, was the value of the average yearly produce,
rating it at 25 cents per bushel, and reckoning the dollar of 100 cents
to be worth 4s,?
4. Required the mean of the following observations of temperature :
—41° 29', 41« 27^', .39° 13', 41° 33', 37° 47^', 44° 28', and 40** 13'.
5. If 3 quarts of stout at 9d, a quart are mixed with 10 pints of ale
at 2jc?. a pint, what is the worth of a pint of the mixture?
^. At a competitive examination Ihere were 4 candidates at the age of
J9, 3 at SO, 2 at 21, »n4 3 at 23. Find \\v^ VJec?^'^ ^^«
EXAMINATION-PAPERS. 185
7» How many square feet are in a regularly tapering plank 10 ft.
6 Id. long, the width being 9 inches at one end and 7 inches at the
other?
8. The average of twenty- one results is 61, that of the first eight
being 64, and of the next eleven 59. Required the average of the last
twa
9. Three quantities of tea, at 3/8, 4/2, and 4/4 per lb., respectively,
make a mixture of 136 lbs., there being 5 lbs. more of the first kind than
of the second, and 6 lbs. more of the third than of the first and second
together. What is the worth of the mixture per lb.?
10. The average of ten results was 17|; that of the first three was
16|,andof the next four 16|; the eighth was 3 less than the ninth, and
4 less than the tenth. What was the last result?
11. If 9 gallons of spirits at 18/6 are mingled with 7 gallons at 21/,
how much water must be added to reduce the value to 16/6 a gallon?
Paper VIII.
Questions on the Relation between Time and Tower,
1. M can do a piece of work in 20 days of 7 hours, and iVcan do it
in 14 days of 8 hours. For how many hours a day should JIf and N be
engaged together, that the work may be done in 10 days?
^does I measure of work per hour;
140 such measures = the whole work.
iVcan do, per hour, the 1 12th of the whole,
viz. 140-^112, or Ij measures;
/. M and iV together do 2\ meas. per hour;
or the whole work in 140-7-2j«62? hrs.
62| hrs. = 10 days, is 6§ hrs. a day. Ans,
2. A cistern is filled by two pipes, A and J9, in 20 and 24 minutes
respectively, and is emptied by a tap C in 30 minutes. What part of it
will be filled in 15 minutes, if A, B, and C are all turned on together?
If A runs 1 measure per minute, 20 measures would fill
the cistern ; then B would run, per minute, the 24th of 20,
5 2
viz. g of a measure, and C the 30th of 20, viz.- of a measure;
and Af B, and C being all opened, the cistern would gain
5 2
1 + - — ^, orlj meas. per minute, and in 15 min. would gain
o o
Ijx 15 = 17j meas.,
7
which is 17^ twentieths =^ T of the cisletn. Am^
1
186 EXAMINATION-PAPERS.
3. F and G together reap a field in 8^ days, and F alone can reap
as much in 3| dajs as G can do in 5. In what time could each by him-<
self reap the field?
1 7
Fin 1 day does 1 measure, G - of Sj mea8.=--.meas.
5 10
/. the whole work is 1^ ^ 85== 14| meas.
14|-T- 1 = 14| da. by F alone T
14|^l=21Jda.byGalone,J
4. Y and Z began together a piece of work which they could have
done singly in 34 and 38 days, respectively. Y continued till the work
was finished; but Z had left him 4 days before its completion. In what
time was the work done?
y did 1 measure per day, and the whole work was 34 measures;
17
so that Z did, per day, the 38th of 34 = — of a measure.
19
17
Now, if Z had continued the whole time of K 4 times -— . or
19'
3j^ extra measures of work would have been done, viz. 37j|
meas. by both agents in Ys time; therefore
37ii-^llJ==714-J-36 = 195 da. Ana.
5. A cistern has two supplying pipes/ ^ and 5, and a tap C. "When
the cistern is empty, A and B are turned on, and it is filled in 4 hours;
then B is shut and C turned on, and the cistern is quite emptied in 40
hours; when, lastly, A is shut and B turned on, and in 60 hours after-
wards the cistern is again filled. In what time could the cistern bo
filled by each of the pipes A and B, singly ?
A and B together supply 1 measure per hour,
and the whole content of the cistern is 4 measures.
B runs, per hour, more than C, — of the 4 meas.
60
1
C runs, per hour, more than A^ — of the 4 meas.
40
4 4 1
A B runs ^ + — , or - meas. per hour more than A»
60 40 6
/. A and B together, in 1 hour, run - meaa. more than A
6
runs in 2 hours;
but A and B together run 1 measure per hour;
.*. A runs 1 — -, or - meas. in 2 hours,
EXAMINATION-PAPERS. 187
5 meas. : 4 meas. :: 2 hrs. : 9| hrs. by A/\
>An8.
I meas. : 4 meas. :: 2 hrs. : 6f hrs. by B,
^6. ^ can do a piece of work in 25 days, B can do it in 20 days,
and C in 24. The three work together for 2 days, and then A and B
leave; but C continues, and, after 8| days, is rejoined by A, who brings
D along with him, and these three finish the remainder of the woik in
3 days more. In what time would D aloHc h.ive done the whole work?
7. A piece of work can be done by A and B together in 14 hours,
or by B and C in 10| hours, or by A and C in 12 hours. In what
time could each person do it by himself?
8. To complete a certain work, B would take twice as long as A and
C together, and C thrice as long as A and B together; and A, B, and
C, by their united exertions can do it in 5 days. In what time could
each do it by himself?
9. A can do a piece of work in 10 days, B in9, C in 12. They
all begin it together ; but only C continues till the work is finished, — A
leaving it 3| days, and B 2| days before its completion. In what time
is it performed?
10. A cistern has two pipes, A and B, which singly could fill it in 9
hours and 10 hours, respectively. It has also two taps, C and D, which
singly could empty it in 12 hours and 8 hours, respectively. Suppose
that when the cistern stands half-full of water, A and D are turned on
for 3 hours; that then B is also turned on for the next 2 hours; that
then A and D are turned off, and C is turned on for the next 8 houi-s;
after which all are shut, and the cistern is found to contain 95 gallons
more than its half content : — Find the content of the cistern. Find also
how much per hour the cistern would lose or gain, if all the pipes were
set open at once.
Paper IX
Questions on Exchange, (See Note XL)
1. BcJuce 396 dollars 53 cents American to British money, at
As, 6d, per dollar.
2. Convert 1206.70 American dollars into French money, at 5 franca
45 centimes per dollar.
3. Reduce £3758 165. 6d. to francs, at 25.35 francs per £.
4. Find the value, in British money, of goods aold iot *1%^^ Ixv&k»
90 ceatmea^-^r^xchange, 24 ^r. 41} ct8. per £«
188 EXAMINATION-PAPERS.
5. What in English money is the value of the franc, at the exchange
of 25.57 francs per £ sterling?
6. How many pence per milree ( = 1000 rccs) is the exchange
between Portugal and Britain, when £823 5s, 6(f. worth of wine costs
8161 milrees 375 rees?
7. If, when the course of exchange between England and Spain is
38j</. per dollar of 20 reals, a merchant in Liverpool draws a bill of
£354 16«. Sd. on Madrid, how many dollars and reals will pay the draft?
8. What is the arbitrated rate of exchange between London and"
Lisbon, when bills on Paris, bouglit in London at 25.65 francs per £,
fire sold in Lisbon at 525 rees per 3 francs?
9. If 11-65 Dutch florins are given for 24*89 francs, 383 florins fur
437 marks Hambro', and 685 marks for 32 silver rubles of Petersburgh;
how many francs should be given for 932 silver rubles?
10. Reckoning a Roman scudo worth 5| francs, and a shilling worth
\\ franc, what amount of discount do I allow by accepting £10 in
exchange for 45 scudi and 12 francs? And if I were to allow 4 per
cent, discount, how many francs along with 50 scudi should I give for
£12?
11. A merchant in London owes to one in Amsterdam 350*75
florins, which must be remitted through Paris. Tlie quotations being,
for London on Paris 25 frar.cs 30 cents, per £, and for Amsterdam on
Paris 45| florins per 100 francs, the London merchant delays remitting
till the rates are 25*45 francs per £, and 11 florins per 24 francs.
What does he gain or lose by the delay?
12. £1000 sterling is due from London to Portugal, when the
exchange is Gl^d. per milree. Whether is it better, for Portugal, to
draw directly on London, or circuitously, at an expense of 1 J per cent.,
through Holland and France; — exchange between Britain and HoUand
11*90 florins per £ sterling, between Holland and France 10 florins for
21 francs, and between France and Portugal 480 rees for 3 francs?
13. Wlicn English money bears a premium of 5 per cent, in America,
how much sterling should be given for 750 dollars, each worth 4«. 6<f. at
par?
14. A rupee contains 16 annas each 12 pice: — Find, in French
money, the annual interest, at 3l per cent., on 5217 rup. 3 an. 6 pi^
exchange 2*63 francs per rupee.
1 5. If goods bought in London at a guinea be exported to Xew
York, at how many dollars should they be sold there, in order to cover
all expenses; estimating the export charges to be 7 J per cent., and the
sale charges 5 per cent. ; the course of exchange being 6 per cent, pre-
jDiam for bills on London?
£XAMlNAT10^'-PArEIt«. I8d
16. At what price in Company's rupees (each = 16 annas) was
indigo purchased in Calcutta, if the •ale of it in London at 5s, per lb.
yielded a profit of 20 per cent.; the shipping charges in Calcutta being
6 per cent., sale charges in Xiondon 9 per cent., and loss of weight 1|
per cent.: — exchange 25d. per rupee?
17. Given — that 1 ounce Troy equals 31*1 grammes; that 10 '
grammes of French standard gold are worth 31 francs; and that the
worth of a given weight of English standard gold is to that of the same
weight of French standard as 3151 to 3100: — i
(i.) To what number of Troy ounces of English standard gold is
the franc equivalent, and what is the fixed number of francs equivalent
to £l? — the English mint piicc for standard gold being 77s. 10^. per
oancc.
(ii.) How many francs are equivalent to £l, when gold purchased in
London at 77«. 1U|(/. is sold in Paris at 14^ per mille (i.e. per 1000)
premium on the fixed price? and how many, when gold is at 1 per mille
.discount?
(iii.) Find that the results are correctly stated in the following
newspaper reports; and give the percentage results more nearly ; —
a. The premium of gold at Paris is 7^ per mille, which, at the
English mint price of £3 \7s. lO^d, per ounce for standard gold, gives
exchange 25*35^; and the exchange at Paris on London, at short,*
being 25*33|, it follows that gold is about 0.09 per cent, dearer in Paris
than in London.
b. The quotation of gold at Paris is about ^ per mille premium, and
the short exchange on London is 25*27^. On comparing these rates
with the English mint price of £3 17 s. lO^J. per ounce for standard
gold, it appears that gold is nearly 4-lOihs per cent, dearer in London
than in Paris.
Paper X.
QueJiona on the uniform consumption of uniformly growing product, ^
1. Suppose that in a meadow of 20 acres the grass grows at a
Uniform rate, and that 133 oxen could consume the whole of the grass
in 13 days, or that 28 of the oxen could eat up 5 acres of it in 16 days;
how many of the oxen could eat up 4 acres of it in 14 days?
133 ox. to 20 ac. is 26| ox. to 4 ac
28 ox. to 5 ac. is 2-2f ox. to 4 ac.
* That !% by bll's parable at short light, as 3 day »' iifcU, anei \.\\aclot^ VaOTi«^\aXAi
Worth thvlr amount In auh.
I
190 EXAMINATION-PAPERS.
16 da.
22'4ox. : 26'6ox.::13da. : 15^ da.
3 days' growth eaten by 22.4 ox. in jo^^
— da. : 16 da. ::3 da. growth : 85§ da. growth.
16
16
.*, the original grass is = 69| da. growth.
69j 69j
16^ li.
85| da. growth : 83i da. growth 1 ^^^^^ -. o^ ^^^
14 da. : 16 da. J
Note. In explanation of the above Torm of solution, it may be obserred that at the
orig. grass+ 13 da. growth of the 4 acres is eaten by 26*6 ox. in 13 da.
.'. orig. grass +13 da. growth is eaten by 22 4 ox. in 15^ da.
but, orig. grass +2fi da. growth is eaten by 22'4 ox. in IC^ jda.
.*. 3 da. growth is eaten by 22*4 ox. in ^ da.,
which amounts to 85^ da. growth in the whole IG da.;
flo that the quantity of grass in the meadow at first must have been G9^ days' growth |
and we have now given, orig. grass+IG da. growth eaten by 22*4 ox. m IG da., to find
how many ox. would eat orig. grass+14 da. growth in 14 da.
For anollier manner of solving problems of this kind see Hunter*s Art qf Teaching
Ant/itnetiCt p. 105, and Examination Questions on ' Colenso's Algebra,' p. G2.
2. If 133 oxen consume the grass of a meadow in 13 days, and
112 of the oxen could consume the grass of the same meadow in
16 days, — the grass growing uniformly; in what time could 125 of the
oxen do it?
Here, as in the preceding solution, the original grass will bo
found = 69i days* growth; and now, 16 + 69| da. growth being
eaten by 112 oxen in 16 da., tlie time is required in which 125
oxen would eat what grows in the required time + 69g dd. growth.
112 : 125 ox. ) .. g.i j^ growth : 5|? da. growth.
16 : 1 da. 3 3 fa 21 o
or, 125 oxen eat 5|j da. growth in 1 day,
1
thus consuming 4|^ da. growth of the orig. grass per da^i
or the whole in 69i-r4|5« 14 da. Ans,
% 3. If 29 oxen would eat up a field of grass in 7 weeks, or 25 oxctt
would eat up the same field in 9 weeks,— the grass growing uniformly^
how many oxen would do it in 6 weeks?
4. Suppose that a tank receircs a regular and continaal supply of
t:XAMlNATION-PAt»EIlS. 191
water, and tbat, when it contains a certain quantity, 12 equal taps being
set open wonld empty it in 7| minutes, or 7 of the same taps would
empty it in 16 minutes; how many of the taps would empty it in 50
mmntes?
5. Suppose that in a certain meadow the grass is of uniform quality
and growth, and that 20 oxen would exhaust the grass in 12| days, or
21 oxen would do so in 12 days; in what time would 26 oxen dg it?
6. I find that I can engage 15 workmen for 11 weeks, or 31 work*
men for 5 weeks, at uniform wages, and in either case pay the wages
exactly by means of the interest now accumulated on a certain sum of
money and that which will arise during the particular period of engage-
ment: — ^For how long could I engage 9 workmen on the same prin-
ciple?
7. If 23 oxen consume 8 acres of pasture in 26 days, and 25 oxen
consume 7 acres of the same in 20 days, — the grass growing uniformly;
how many acres of it would 33 oxen consume in 5 J days?
8. Suppose that 17 oxen in 30 days, or 19 oxen in 24 days, could
consume a field of uniformly growing pasture; find what number of
oxen, diminished by the removal of 4 at the end of 6 days, would eat up
the same field in 8 days.
9. In a field in which grass grows uniformly, suppose that 31 oxen
can consume 8 J acres in | of the time in which 15 oxen would consume
5\ acres, and that 22 oxen would require 3 days longer to consume 7|
acres than 20 oxen would require for 6j acres: — ^In what time would
the 31 oxen eat up the 8| acres?
10. An empty cistern has two supplying pipes A and B, and two
taps C and Z>. A would fill the cistern in 42 J minutes, and ^ in 46
minutes; and D can carry off" per minute half as much again as C,
After A and B, running together, have supplied a certain quantity, C is
allowed to run with them, and takes 51 minutes to empty the cistern;
but had J> been turned on along with C, the two would have taken only
5j minutes to empty it. In what time would the cistern have been
emptied if J) had been turned on instead of C ? and how much of the
cistern was filled when C was set open?
9
m
Paper XI.
Questions similar to Concluding Misc. Examp, 134 §• 194.
1. A certain number is divided into two parts, such that 10 times the
first added to 18 times the second gives 15 times the entire number;
what fraction of the whole is each of the parts?
Questions of this kind closely resemble £Jxamp. ^ V^
Paper YL, and may be solved simUaily, toa^ waRfe'^^^iK^^
02
192 EiAMlNATION-PAI»EItS.
10 times the first part and 18 times the second together equal
to 15 times the first and 15 times the second, it is evident that
(15 — 10) times the first compensates or equals (18 — 15) times
the second ; i.e. 5 of the 1st ^3 of the Snrf; or, 1 of the lst=
3 "3
- of the 2nd', or, Ist : 2nd :: 3 : 5; so that the paits are -
5 8
and - of the whole.
8
Otherwise,
10 times the l^^ with 18 times the 2n(f»15 times both;
10 10 10
.*. 8 times the 'lnd=^ 5 times both;
or, the 2nd is - of the whole
8
o
and the \8ti% - of do.
8
2. Divide the quantity 520 into two parts, such that 118 times one
part added to 128 times the other shall give 63700.
Here, we have 63700 [-f-520] = 122| times the entire no.
.*. 118 times the \st with 128 times the 2nd—\22\ times both;
/. 10 times the 2«d= 4^ times both;
Q
or, the 2nd is ~ of the whole, = 234
20
and the Ist is — of do. ^
20
*-Ans,
= 286
3. A person borrows £618 in two separate sums, at the re&Vective
rates of 3| and 5 per cent, per annum; and he repays the two loans at
the end of 10 months, with interest amounting to £21 10s. Kcquired
the umouut of each loan.
The respective interests are
5 of -?^ of I«i loan, and i of -A of 2ndi
6 100 6 100
22- 15
and these together are equal to —^ or -^ of both loans.
618 412
i.e. 2- of \st with — of 2nd^~ of bo:h;
240 240 412
• t
)
• --of2nJ=/^Ii 1 .
240 V412 240^
716
oi2nd=^( — -\ of Ictb
V4i:
412 x240
j\ 2;je/=-^l^of je618ȣ358l ^
412x3 \^Ana,
of £618.
EXAMINATION-PAPERS. 193
% 4. Sold 449 yards of cloth, part at 12*. a yard, and the reminder
at 17«., and for the whole received £315 13*. How many yards were
sold at each rate?
5. A woman sold 7J dozen apples for 6s. 2d,, some at the rate of
8 for 2|</., and the rest at 8 for 6^d How many were sold at each
rate?
6. I gave 85. for a basket of oranges and lemons, buying the former
at the rate of 2 for Sd„ and the latter at 5 for 4d, I then sold all at tlie
uniform rate of 5 for 6(f., and gained 6| per cent. How many had I
of each kind?
7. 12 lbs. of tea and 25 lbs. of coffee together cost £4 6«. Sd.; but if
tea were to rise 2^ per cent, and coffee to fall 4| per cent., the same
quantities would cost £4 5*. lid, Kequired the pri'^es of tea and coffee
per lb.
8. If the increase in the number of male and female criminals be 1*8
per cent., while the decrease in the number of males alone is 4*6 per
cent, and the increase in the number of females is 9*8; compare the
antecedent numbers of male and female criminals.
Paper XH.
Questions on Involution and Evolu'.ion,
1- Simplify the expression - of — x v/ -•
To remove surd dcnomirators, multiply the numerator and denomin-
ator of the second fraction by * 5, and those of the third fraction by
-v/3, which gives
3 . 300 y a/5 a/6 60 ,^^ .
- 01 X ^ — = - \/30. Ans,
7 6 3 7
2 Which is the greater quantity, a.'2 or V3?
1 13 3 1 X
2^ and 33=2« and 3^ = 8« and 9";
/. ^3 is the greater.
3. Find the diagonal of a rectangular space, 792 feet long and 406
feet broad.
The length and breadth form with the diagonal a right-
angled triangle, of which the two perpendicular sides are
given, to find the third or longest side. Now, in every right-
angled triangle, the sum of the squares of the perpendicular
Bides is equal to the square of the longest side; therefore,
792* + 406*=792100, square of diag,
a/792100^S90 ft., the diagoi\a\. Ana.
194
EXAMIKATION'FAFERS.
D
^V •
\.
\. f
^
^^ •
^^ •
\,
•
f
»
1
»
/
c
\.
/ /V
^v^
' '' >v
>v
;'' \
A
B
4. Show tbat the length of the edge of a cube multiplied by VS
gives the diagonal of the cube.
If AB and Bc, edges of a cube, be each
represented by 1, then the square of AO, the
diagonal of a snperRcial side, is evidently
1*+ 1^=2, and the square of the cube's dia-
gonal AD is = AC* -f CD^ = 2 + 1 1= 3 ; therefore
AD = VS when the edge of the cube is 1 ; or,
by similar triangles, the diagonal of every
cube is the product of the length of the edge by v/S.
5. The tip of a reed was 8 inches above the surface of a lake; but,
forced by the wind, it gradually advanced, and was
submerged at a distance of 28 in. Find the depth of
the water.
Let AD=DC represent the reed, bc the
surface, bd the depth ; ab = 8, bc = 28 ; to
find BD. Draw ac, and de bisecting it.
AC« = 64 + 784 = 8i8; AB«=iAc2=:212.
Twice the area of adc == db x ac = bc x ad ;
AD«
AC*
DE*
BC*
AD*-AE* , AD* AD*-212
BC*
or =
818
784 '
or 63(ad*-53x4) = 49ad*; /. ad*x4
•r 53* X 4 ; hence ad = 63 ; .*. bd = 63 - 8 = 45 inches. Ans.
Otherwise : In the triangle dbc we have bc = 28, and the
difference of dc and db=s8 ; .•.28*-^8 = 98, the sum of dc and
db; and hence J(98 — 8) = 46 in. Ans, (/Sec Note, art. 108.)
6. What quantity is — of its reciprocal?
The reciprocal of any quantity wi is - ; and here m = 'SSQA
m
8696, and Wr= ^-8696 = -9325. Ans.
m
m*
^ 7. A square space contains 1056 sq. yards: Express the length
of its side as the decimal of ^ of a mile.
8. Find the side of a square field containing 2 ac. 3 ro. 17 po.
80 yds.
9. A square space contains 38 sq. poles 6 yds. 4 ft. 72 in.; find
the length of its 8ide«
10. A rectangular field is 190 yds. long and 123 yds. wide; find the
side of a square field of half the area; find also the length of a field
twice as large as the first, and twice asVm^ «a \t\& btQa4«
11. Show that 10-5- ^/2 i8=5 X V2.
EXAMINATI0X-PAPER8. 195
12. Multiplj 'v/n2 by -v/175.
13. If the perpendicalar sides of a right-angled triangle arc 13*02
and 5*2 feet, what is the third side?
14. If the town ^ is 72 miles west of B and 135 south of C, wliat
is the distance from B to C?
15. Which is the greater of the two quantities ^9 and t/l9? an<?
which of the two -v/3 and -^15?
16. If the diagonal of a rectangular surface is 3*4061 inches, and
the length 3*406 inches, what is the width ?
17. The diagonal of a square is 353*55; find the length of its sido^
18.' The members of a party being solicited for contributions to a
charitable object, each person gave a number of half-pennies equal to
the number of members, and thus made up a sum total of I2s, Old,
What sum was contributed by each ?
19. Suppose the top of a straight ladder, 18| feet long, to rest
against a building at the height of 13| feet from the ground; at what
horizontal distance from the bottom of the building is the foot of the
ladder placed?
20. The edge of a cube is 250; what is its diagonal?
21. Find the edge, and also the surface, of a cube of wood, the
diagonal of which is 3 ft. 9 in.
22. Of what sum of money is £28 the same fraction that the sum
itself is of 60 guineas ?
23. If the compound interest of £250 for 2 years be £20 8«., what
is the rate per cent, per annum ?
24. The capacity of a cistern is 478*4 gallons :— Required (a) the
length equal to the breadth of a cistcra of the same capacity 2| feet
deep ; and (6) the breadth equal to twice the depth of a cistern of the
game capacity 6 feet long:— a gallon being = 277*274 cub. inches.
25. What fraction of ( 'v/4050 x -002 -r '20 + -v/HSS) -r- ^02 ia
>v/(6*OOS-f-*3042)+ ^/(llG'6 x -OlG)?
^6, A can excavate 14*2884 cubic yards per day; how many can B
do per day, if A could do B^s daily quantity in - of the time that B
would take to do A*8 daily quantity ?
27. The original cost of a pipe of port is £55, and it is sold to A at
a certain loss per cent.; then A sells it to B at the same losing rate; but
B sells it to C, at a profit of 12 per cent., for the original cost. What
was the loss per cent, at which A and B sojd the wine?
196 EXAMINATIOW-PAPERS.
Paper XIII.
Supplementary Miscellaneous Questions, [A.]
1. What is the greatest unit of time with which 15 ho. 12 min. and
1 da. 3 hr. 33 min. can be both represented by integers?
2. How many times can '0087 be subtracted from 2*291, and what
will the remainder be?
3. What is the greatest number by which 2500 and 3300 can he
divided, so aa to leave remainders 4 and 36, respectively ?
4. Define Proportion. — Can the quantities 2 yds. 2 ft. lOj in.,
£24 3s., £12 lis. 6j</., and 5 yds. 2 ft., be formed into a pioportion?
Give the reason.
5. State the distinction (i) between simple and compound division,
(ii) between simple and compound proportion^ and (iii) between simple
and compound interest.
6. Distinguish mercantile from true discount; and show that the
difference between the interest and the true discount on the same sura
is the interest of the discount.
7. Find by duodecimal multiplication the product of 13 ft. 5 in.
7 pts. by 3 ft. 5 in,
8. Multiply, by the method of duodecimals, 29 ft. 7 in. by 9 ft.
6 in. 6 pts.
9. Express the results of the two preceding questions in squaro
feet, square inches, and a fraction of a square inch.
10. Find, by duodecimal multiplication, that the product of 26 ft.
8 in. by 5 in. 9 pts. is 12 sq. ft. 9' 4'' ; and calculate by Practice the
value of the latter quantity at 15«. 9j</. per square foot.
11. What two quantities have for their sum 9 guineas and 9 shil-
lings, and for their difference 10 crowns and 10 pence?
12. A offers to 5 6 cwt. 2 qrs. 7 lbs. of sugar, worth 38». per cwt.,
for 24 yds. of cloth, worth 8s, 3f(/. per yard. How much per cent,
would B gain or lose by accepting the offer?
13. If one man can plough a quarter of an acre in 2 hrs. 23 min.,
and another can do it in 2 hrs. 34 min., wl^at fraction of an acre could
they together plough in an hour?
2 4 7
14. What sum of money increased by 3 of - of - of itself amounts
5 5 8
to 3s. 4d.?
15. What decimal fraction diminished by '037 of itself becomes
•6955?
16. Show that the amount of £7 for 3 years, at 5 per cent per
annum, compound interest, is-i^"* xVO^\
EXAMINATION-PAPERS. 197
17. If 3f per cent, is lost hj selling steel nibs at Ss. QeL a gross, how
much would be gained or lost per cent, by selling them at 2«. 6|(/. a
hundred ?
18. A fruiterer by selling apples at the rate of 8 for 6|J. gains 17
per cent. ; at what rate should he sell them per dozen to gain 20 per
cent ?
19. If by selling cloth at 28*. 6<f. for 5 yards my gain would be 6|
per cent., what should I gain or lose per cent, by selling it at 37«. 6//.
for 7 yards?
20. The population of a town is 3370; what was its population a
year ago, if in the interval there has been an increase of about
2'65 per cent. ?
21. The amounts £210 and £155 are payable 2 years and 5 years
hence, respectively ; assign the mean period, or equated time, at the end
of which, according to mercantile discount, these two amounts might be
paid at once ?
22. The sum of £434 is due as follows :~f' ot'it in 4 months, | in
5 months, and the remainder in 7 months. Find the equated time
for one payment of £434, according to mercantile discount.
23. Find the value of
^-1 of^Ul!J^ofll^I^l!lMJl-of 13duys3hrs.
-If of 44 £2 17*. 2 yds. 1.7 lu
Invent a question to which the last three factors in this expression may
be the answer; and show how they are so.
24. Divide 99 into four parts, so that the first shall contain 3 for
every 4 in the third and every 5 in the fourth, and so that J of the second
may be | of the sum of all the rest.
25. Divide Ss, among -4, B, C, so that A may receive Sd. as often
as B receives 3c/., and B may receive 5d. as often as C receives 3d.
26. Express in lowest terms the product of
»7
6
9 25 49 81 11 59 181
27. The sum of 7|(i. was divided among A^ B, C, in such proportion
that A received 1|</. more than C, and B 2\d. less than C: Suppose a
sovereign had been divided among them in the same proportion, what
would each have received?
28. What half-yearly dividend is derived from an investment of
£1000 in the 3 per cents, at 87 J, after deducting for income-tax 7rf. in
ihe£?
29. What interest does a person obtain for his money, who invests
in the 3| per cents, at 91?
so. How many acres, roods, &c. are cquaX lo - ol ^^o\ ^:^«^^^
198 EXAMINATION-PAPERS.
IS Q^ £3 5s. ^^ 1 lb. 4 oz. 17 dwt 12 grs. ^^ 77 da. 4 ho. 30 m. ^^
•0^6 14«. 3d. 2 lbs. 2 oz. (avoird.) 6 da. 12 ho.
518 sq. ft. 28 in. ?
31. Find the true discount on £100 10*. lOd. payable in 4 years, in-
terest being at 3| per cent, per annum.
32. What sum of money improved by simple interest, at 3| per cent,
per annum, for half a year, will amount to £14 16». ?
33. What would be the true present worth of £294 2*. 6rf., for 3^
years, reckoning simple interest at the yeariy rate of 4.027 guineas per
£100?
34. If the simple interest of £162*871 for 148 days were £2*81 42
what would be the rate per cent, per annum?
Paper XIV.
Supplementary Miscellaneous Questions, [B.]
1. Two numbers have for their greatest common measure 537 and
for their least common multiple 18795. What must the greater n^ be,
if the less is = 105 times ?! of 25521?
4g 8-4
2. The circumference of the fore wheel of a carriage is 6| feet,
and that of the hind wheel is 12f feet. How many feet must the car-
riage pass over before both wheels shall have made a complete number of
revolutions?
3. The diameter of the fore wheel of a carriage is | of that of the
hind wheel, and the former makes 528 revolutions in passing over f of
a mile. How many revolutions does the hind wheel make in passing
over a mile? and what is the circumference of each wheel?
4. In what proportion must water be mingled with spirits worth
10s. 6 J, a gallon, to reduce the value to 9*. lid. per gallon ? ^
1 7
5. How much ore must ono raise, that on losing _i in roesting
40
and — of the residue in smelting, there may result 506 tons of pure
metal?
6. £225 95. is due in 48 days, and £599 Ss. in 26 days: — ^What
sum paid at present would discharge both these debts? and how many
days would be the equated time for one payment of the £824 17*.? —
interest being reckoned at 5 per cent.
/, A cubic foot of water weighs 1000 oz. avoirdupois ; a pipe
wAose bore is 3^ square inches dischaigea ^S^\\i^.^tx mluute; find Ae
reJocitfper hour of the issuing water.
EXAMINATION-PAPERS. 199
8. If when corn is 15*. 9d, a quarter, and hay 5|J. per stone,
7 horses can be kept 8 days for £4 U, Sd.; how many weeks can 16
horses be kept for £95, when corn is 2*. a bushel, and hay 70«. a ton,
supposing that 126 lbs. of hay are consumed with 1 bushel of com?
9. An analysis of the Board of Trade returns for 1861, respecting
shipwrecked lives, gave the following results: — Saved by life-boats, 13j
per cent. ; by rocket and mortar apparatus, 8 per cent.; by ships* boats,
&c., 62 per cent.; by individual exertion | per cent: lost, 16 per cent
I Determine the nnmber of lives saved, by the severaf means enumerated,
corresponding to an excess of 2619 rescues by ships' boats over those
by life-boats.
10. Find two decimal fractions together equal to — , and such that
one may be — of the other.
15
11. A stationer by selling quills at a guinea a thousand, gained f of
what they cost him. What was the prime cost?
12. A ring weighs 1 dwt 4 grs., and is worth £l 2«. If 1050 of
such rings be packed in a box weighing 3| lbs., what would it cost to
convey them 144 miles, at the rate of 5«. per ton per mile, insurance
being demanded at the rate of | per cent. ?
13. A monolith of red granite in the Isle of Mull is said to be about
108 feet in length, and to have an average transverse section of 113
gqnare feet. If shaped for an obelisk, it would probably lose one-third
of its bulk, and then weigh about 600 tons. Determine the number of
cubic yards in such an obelisk, and the weight in pounds of a cubic foot
"Of granite.
14. Show that, in comparing the rates of two locomotive bodies, A
and B, if the distance passed over per unit of time by ^1 is | of that by
B, then A*a time per unit of distance is | of B\
15. A has 88 florins and a sovereign; B has 61 half-sovereigns and
11 florins. What sum transferred by J? to -4 would make B have
exactly 6 times as much money as Af
16. The difference of two numbers is 477^, and one of them is to
the other as ^ of 2| of 1*53 is to 5iJ x 4j. Find the two numbers.
17. With what capital did a tradesman commence btusiness, if at the
end of 12 months his nett gain amounted to £2iu 14^.; a certain
portion only of that gain being accounted trade profit, the remainder,
viz. 5 shillings for every 9 shillings of the trade profit, being legal
interest of capital ?
18. The snm of £l00ha8heea accamnlatlnf^ b^. qotq^oxjccv^Vq^^^^^'^
200 EXAMINATION-PAPERS.
for 125 years at 3 per cent. : the amoant is now invested in 3 per cent,
consols at 95. What will be the annual income therefrom ?
N. B. 103*«= 4*383906; and only four places of decimals
need be retained in the result.
19. If the discount on £567 be £34 I4s. 3f(f., simple interest being
reckoned at 4| per cent., when is the sum due?
20. A narrow rectangular field, ABCD, has its length AB 160
yds. and breadth BC 3 If yards. To what point E in the side AB must
a straight line from C be drawn, so that AECD may contain an acre?
21. A person invests £6200 in the 3 per cents, at 89|, and pays
income-tax lOJ. in the pound; on the stock rising to 92 he sells out,
and invests the proceeds in £50 railway shares which yield an annual
dividend of Z\ per cent , clear of income-tax. Find the alteration in
his income.
22. Certain railway shares pay an annual dividend of £3 IO5. A
person having bought 12 shares, at such a price that they yielded 5| per
cent, on his investment, sold them when the price had risen £5, and
invested the proceeds in 3^ per cent, stock at 85. Find the alteration
in his income.
23. What fraction of ^01 35 is ^004.
84. From — of V5-92 subtract -1 of V6177.
27 b7
Paper XV.
Suppkmeniary Miscellaneous Questions, [C]
1. A corn merchant having bought 1300 quarters of wheat, sold
oncfiftli of it at a profit of 5 per cent., one-third at a profit of 8 per
cent., and the remainder nt a profit of 12 per cent.; but had he sold all
at a profit of 10 per cent., his gain would have been £16 13«. Sd, more.
What did the wheat cost him ?
117
1 — — = - sold at 12 p. c. profit.
5 o 15
,*, the several quantities are as 3, 5, and 7.
£3x105 = £31 5
6x]08= 6-40
_7xl-12« 7-84
16-39
15x110= 16-50
EXAMlNATION-PAPEnS. 201
That is, on every £15 of the whole prinle co:t the gain
would have been £.11 more ; hence,
£ 11 : £16 135. sd. :: £15 : £2275. Ans.
Hi The gross receipts of a railway company in a certain year are
apportioned thus : — 40 per cent, to pay the working expenses, 54 i)cr
cent to give the shareholders a dividend at the rate of 3| per cent. 0:1
their shares; and the remainder, £28350, is reserved. Find the paid-up
capital of the company.
100-40-54= 6 p. c. of gross receipts is reserved,
/. 6 : 54 :: £28350 : £255150 amt. of dividends.
3^ : 255150 :: £100 : £7290000. Ans,
3. What is the exact time between 5 and 6 o'clock vvhcn the hour
and mmute hands of a watch should be at right angles to each other ?
and what, wheil they should be coincident?
Call the hour hand H, and the minute hand M. At
5 o'clock, H is 5 twelfths of the circumference in advance of
M; and it is required to find at what time after 5 o'clock the
interval between H and M will be 3 twelfths.
Now, as (5 — 3) twelfths and (5 + 3) twelfths are both proper
fractions, there WlU be two occurrences of the interval.
In the first instance, M has to gain 2 twelfths on H, and in
the second instance 8 twelfths-, and, as M goes 12 times as
fast as H, and gains 11 twelfths of the circumference per hour,
we have
11 tw. : 2 tw. :: 60 min. : lO^ min. past 5;
11 tw. : 8 tw. :: 60 min. : 43^ min. past 5;
which are the times when the hands intercept a fourth of the
circumference, or are at right angles.
Similarly, to find when the hands are coincident is to find
when M will have gained 5 twelfths of the circumf. on //.
11 tw. : 5 tw. :: 60 min. : 27^ min. past 5;
^hich is the time when Hand 3f point in one direction.
Note, The third answer might have been found thus:
(10i? + 43^)^2=27i\min. past 5.
4. At what rate must I sell sherry that cost me 40«. a dozen, if I
am to gain on every £100 of outlay the selling price of 5 dozen?
£l00-^£2 = 50 dozen bought for £100;
and I am to sell (50—5) or 45 dozen for the prime cost of
50 dozen, viz. for £100 ;
/. £l00-i-45=44*. 5^. per doz. Ans.
5. A*B present age is to B*a as 9 to 7 ; and ^4 "j^w^ «i^o ^^ V^>"
portion was 5 to 2, Find the present age o£ each.
202 EXAMIKATION-PAPBRS.
la solving such problems it is borne in mind that tBe
difference of the ages of two persons is always the same, though
the ratio of the ages is always varying.
Here, then, we have As present age to JS's as 9 : 7 ; and 9
is 4^ times (9 — 7). Similarly, A^a former age was to B*b as
5:2; and 6 is 1| times (5-2).
Therefore, A*a present age is 4^ times the difference of As
and B*s ages ; and his former age was 1| times the same dif-
ference ; so that we have
^'s former age =-| or — , of his present age }
17
/. -^ of As present age = 34
/. As present age = 54/
-B's-of54, =42.
9
Ana,
6. A boatman fows 5 miles with the tide in the time he would take
to row 3 miles against it ; but if the hourly velocity of the current were
^ a mile more, he would move twice as rapidly with the tide as against
it. What is his power of rowing in still water ?
If 5 represent his rate with the tide, then 3 represents bis
rate against the tide, and the average of these, viz. l{5 + 3), or
4, represents his rate in still water; also 5 — 4, or 4— 3, viz. I,
represents the velocity of the current, = ^ of his rate in still
water.
Again, if 2 be his rate with the tide, and 1 his rate against
it, then ^(2 + 1), or 1 ^, is his rate in still water ; also 2 — l^, or
1^ — 1, viz. i, is the velocity of the current, = | of his rate in
still water.
/. _ — _, or — of his rate in still water is = i a mile per
hour ; and hence his rate in still water is } a mile x 12 = 6 mit
an hour. Ans.
7. A contractor engages what he considers a sufficient number of
men to execute a piece of work in 84 da^s ; but he ascertains that three
of his men do, respectively, -^ -, and -, less than an average day's
work, and two others- and — more ; and In order to complete the
8 10 ^
work in the 14 weeks, he procures the help of 17 additional men for the
S4tb day. How much less or more than an average day's work on the
part of these 17 men is required t
EXAMINATION-PAPERS. 203
Here, instead of 5 men working with ordinary ability,
during tho 84 days, there arc
I + f + 1 + 1 + fj =11215 ordinary men;
go that the deficiency to be made up is equal to the irork of 1
J no
ordinary workman for 84 times — -- da,
^ 2520
493
■B 1 ordinary workman for — days,
29
e»17 ordinary workmen for — of a day,
or, 17 men each doing — less than an average day*s work. Am,
8. A farmer gave for a horse a bill of £73 dae in 1 month, and sold
him at once for a bill of j£87 at 4 months. Bcquired the farmer's gain
per cent., reckoning interest at 4| per cent.
lOOf : 100 :: £73 : £^, Pres, Worth of £73 1
lOlJ : 100 : : £87 : £^^, Do. of £87}
7
.'. 822:6io..ioo:ioox«xii=ii7?i
11 7 7 8
or, 1 7f per cent. gain. Ans,
9. Divide the number 237 into three parts such that 3 times tho
first may be equal to 5 times the second and to 8 times the third.
Since 5 times the 2nd =3 times the 1st,
/, the2nd=|ofthelst;
5
5
similarly, the 3rd=:l of the 2nd}
8
3 5 3
and the three parts are as 1, -, and - of -^
5 8 5
or, as 40, 24, and 15;
/. 1? of 237 = 120, the 1st,
79 '
- of 237 « 72, the 2nd,
l? of 237 « 45, the 3rd.
79
Am.
la Divide £5433 I8& into three sums, snch that their amounts by
tompound interest at 5 per cent, per annum, for 20, 23, and 27 "^^vc:^
respectlTely, shall be equal.
204 EXAMINATlOK-PAl»£RS«
The 1st X l-052» = the 3rd x 105",
/. the l8t = the 3rd x 106' ;
The 2nd x l-05'» = lhe 3rd x I'OS^',
/. the 2nd = the 3rd x 1-05*.
Thus, the three required parts of the given sum will be &%
100', 105*, and 1 ; or, as 1-4071, 1-2155, anJ 1 ;
or as 14071, 12155, and 10000 ;
accordingly, the 36226th part of the given sum, viz. 3.?., ninl*
tiplied by these proportional numbers gives £2100 13«.,
£1823 5«., and £1500. Ans.
^11. Suppose 9 men or 15 women to earn 25«. a daj at reaping,
when they work 9^7 hours a day; how many men with 4 women would
earn S5s. a day at the same employment, if the duration of daily work
were an eighth less than in the former case ?
12. Thirteen horses do the same work as twenty ponies, and 12
horses can just draw a certain load on level ground; how many ponies
along with 5 horses could draw a load - as heavy up a gradual slope
which makes the traction more laborious by - for ascent and — fur
•^8 10
roughness?
13. What must a person have invested in the 3 per cents, at 90|, if a
transfer of - of his capital to the 4 per cents, at 115 would increase his
income by £7?
14. Suppose that from an official return of the arrivals of oxen,
calves, sheep, pigs, and horses, in the port of London, from the conti*
ncnt, in a certain week, it appears that there were 3 times as many sheep
ns oxen, that the number of pigs was 13^ per cent, of the number of
sheep, that for every 28 pigs there were 25 calves, that the horses were
- per cent, of the whole, and that the horses and oxen together were
3587: — What was the number of oxen?
15. A merchant has three qualities of whisky, viz. at 18«., 16^., and
lbs, a gallon, and in quantities, respectively, as 3, 4, 5 ; and with these
he mingles such a quantity of water as makes the average value
15jf. 6d, a gallon. How much per cent, of the mixture is water?
1 6. Suppose that 1 5 men would be necessary to excavate 966 cubic
yards in 8 days of 10| hours each:— How many men did a contractor
engage for 12 days of 7^ hours, to excavate 575 cubic yards, if he.
found it requisite to engage 4 additional men during the last 4 days^
in order to complete the work in the 12 days?
for lOj months, and G receives ^ of the gain. Kequired G*8 period of
EXAMINATIOlJ-PAPERS. 205
17. I bought 128 yards of cloth for £100, and am now ohliged to
sell it at a loss of as much money as I shall receive for a dozen yards.
At wliat do I sell it per yard ?
18. I bought paper at the rate of 3«. 7id. for 5 quires, and sold it so
ag to gain as much on the cost of 32 quires as 3 quires were sold for.
At what rate did I sell it per quire ?
19. I gave 3 sovereigns for two dozen of wine, at different rates per
dozen ; and by selling the cheaper kind at a profit of 15 per cent., and
the dearer at a loss of 8 per cent., I obtained a uniform price for both.
A\ bat did each dozen cost mc?
SO. F and G are partners in trade; F contributes ~ of the joint capital
5
5
8
investment.
21. At what time between 11 and 12 o'clock will the hour and
minute hands of a clock make with each other an angle intercepting 27
of the minute divisions?
22. A merchant buys two pipes of wine, one for £1 12, one for £120,
and he also buys a third pipe; on mixing the three, be sells his wine at
50s, per dozen, gaining 25 per cent, on his outlay; what was the price
of the third pipe? — The n° of dozens in a pipe is 56.
23. My age is 62, and my son*s age 30 ; how long ago was my age
5 times that of my son ? and how many years hence (if we are both
alive) will my age be a third of 5 times his age?
24. My age was 24 when my eldest son was bom, and when I attain
to twice my present age he will be 8 times as old as he is now. What is
his age?
25. A boatman rowing against the tide passes a body floating with
the tide, and in 9 minutes afterwards is a mile distant from it; in 35
minutes more he rows 2| miles, and then returns. At what rnte per
hoar does he return, supposing the tide to flow uniformly in one direc-
tion?
26. A com merchant bought 121 quarters of wheat, and he sells it
80 as to gain 17| per cent, on 26 quarters, and 13 per cent, on the
remaining quantity, having previously tried to sell the whole at a uniform
advance of 15 per cent., which would have brought him £4 5s. more
than he actually received. What did the wheat cost him per quarter?
27. A watch that gains 24 seconds per hour is set to right time at a
quarter to 5 p.m. What will be the right time between 8 and 9 o'clock
the same evening, when the hour and minute hands o^ X\k^^«L\r\LY3«&
in exactly opposite directions?
P
206 EXAMINATION-PAPERS.
28. Of the whole cost of constructing a railway, f is held in shares*
and the remainder, £400000, was borrowed on mortgage at 5 per cent.
Find what amonnt of gross annual receipts,— of which 40 ]>er cent
will be required for the working expenses of the line, and 8 per cent for
a reserve fund, — will yield to the shareholders a dividend of 4 J per cent,
on their investments?
29. A dealer buys 18 cwt. 3 qrs. at 1*. 3rf. a lb., which, to obtain a
fair profit, he should retail at 8| per cent, above cost price. But, while
he professes to sell at the rate of 3 lbs. for Ss. \Od.y he serves his cus-
tomers, to his own advantage, with a false balance, in which 10 lb.
weighs 10| lb., and at the same time he uses a false lb. of 6860 grains.
How much does he make beyond the fair profit?
30. I have this day paid £2180, being repayment, with interest, of
two loans, both contracted by me at one time, viz. of £1163 borrowed
at 4 per cent, per annum, and £994 at 4| per cent. How long is it
since the sums were borrowed ?
31. A person borrowed £272 6*. 6d, at 5 per cent per annum, and
repaid the loan by yearly instalments of £100, that sum including the
year's interest; how much of the debt was discharged in 3 years?
32. What must be the gross rental of an estate, so that, after deduct-
ing 7d, in the £ income-tax, and 4| per cent on the remainder for
expenses of collecting, there may be left a nett rental of £1000 ?
33. I sold an amount of railway stock at 104, and invested the pro-
ceeds in the 3 per cents, at 91 ; I then sold out the 3 per cent, stock at
95, and re-purchasing the railway stock at 105, 1 found myself a gainer
of £50 by the whole transaction. Kequu^d the amount of railway
stock.
34. The interest on a certain sum of money for 2 years is
£71 165. 7|<f., and the discount on the same sum, for the same time, is
£63 17*., simple interest being reckoned in both cases. Find the rate
per cent per annum, and the sum.
35. At what rate per cent, per annum, compound interest, would a
sum of money in 2 years amount to the same as at 3| per cent per
annum simple interest?
85. If a publisher, in selling a book for cash, rates it at 25 per cent
below publishing price, and then charges for IS copies as 12, how long
credit could he allow, so that, on the principle of true discount at 4 per
cent per annum, the sum to be received for a book should be just 29 per
cent below publishing price?
37t, The external length, breadth and height of a rectangular woodao
EXAMINATION-PAPERS. 207
closed box are 18, 10, and 6, inches, rcFpcctivclj', and the thickness of
the wood is half an inch. When the box is empty it weighs 15 lbs.,
and when filled with sand, 100 lbs. Compare the weights of equal
bulks of wood and sand.
38. I bought goods at 23s, Od, with 4 months* eredit, and sold them
forthwith at 25*. 6(/. with such allowance of credit as made my gain 6|
per cent. How long credit did I give, reckoning interest at 4 per cant,
per annum ?
39. If I am allowed 1| per cent, discount on an amount charged to
me for goods, and give my acceptance at five months for the nett sum;
and if by selling the goods forthwith for a bill of £l 62 12*. 2c?., payable
in 7 months, my present gain is 11 1 per cent.; what is the amount
originally charged to me, interest being reckoned at 5 per cent, per
annum?
40. The present income of a railway company would justify a
dividend of 4 per cent., if there were no preference shares; but as
£200000 of the stock consists of such shares, which are guaranteed 5
per cent, per annum, the ordinary shareholders receive only 3i per cent.
What is the whole amount of stock?
41. A man bought a house, which cost him 4 per cent, upon the
purchase money to put into repair; it then stood empty for a year, during
which time he reckoned he was losing 5 per cent, upon his total outlay.
He then sold it again for £1192, by which means he gained 10 per cent.
Hpon the original purchase-money. What did he give for the house?
42. (a) Show that if 5 times A, 6 times ^, and 7J times C, are equal
11 2
quantities, then A^ 5, and Care in the proportion of -, -, and -j.
5 6 Id
(b) A charges for a certain article 4 per cent, less than B; how
much per cent, does B charge more than A? If they gain at the same
rate, and the article costs A 305., how much does it cost B ?
43. Divide 33 cwt. 2 qr 22 lb. intothrQs such parts that 6 times th«
first, 9 times the second, and 10 times the third may be eq^iial ai^ounts.
44. Divide £36 Ss, into four parts such tJiat their simple interests for
4, 6, 7, and 10 month*, and at 3, 4, 5, and 6 pej" cent, per annum,
respectively, shall be all equal.
45. Dlvfde £3010 into three sums, so that if the first be pur out at
simple interest for 3 years at 4 per cent., the second for 5 years at 3 per
cent., and the third for 2 years at 2i per cent., the amount of the second
shall be double that of the first, and the amount of the third treble that
of the second.
46. By the sale of goods which cost me £3 19s. ^d.lV'osX. «.«vxcsv
P2
208 EXAMlXATION-PArERS.
equal to 5j per cent, of the proceeds; and by the sale of another
quantity which cost me £o I gained a sum equal to 31? per cent, of the
proceeds. What did I gain per cent, on the whole ?
47. If 9 oxen are kept for the same money as 7 horses (for any given
lime), and a team of oxen arc - as long again in ploughing 97 acres
as the same number of horses are in ploughing 90 acres, and a field
costs as much whether ploughed by oxen or horses, viz. £7 5*. 6^. ; the
same men being required in both cases, and being paid by the time, what
is due to them?
48. If 28 men can excavate 750 cubic yards in 4 days, working C|
hours a day ; what uniform length of day will 24 men require, to
excavate 615 cubic yards in 3i days, supposing that any 5 of the latter
party can do as much in 4 hours as any 6 of the former can do in dj
hour?, and that 2 men will be withdrawn from the latter party- after
2\ days* work ?
49. In a certain manufactory, 158 men of ordinary ability, and
working the same number of hours each day, execute a certain piece of
work in a week ; but if the abilities of 2 of them had been, retpectivelyy
11 3 3
- and - less than ordinary, and the abilities of 2 others . and - more,
7 9 ^ 5 8
23
the work could have been finished '- of an hour sooner. How many
83
hours a day did the men work ?
60. The interval between the firing of two guns, at a railway station,
was 6 minutes, and a passenger in a train, approaching the station at a
uniform rate, heard the second leport 5 min. 51 sec. after hearing the
first Now, suppose the sound of the train's approach to have beoooie
audible at the station when the train was 2 miles off, how soon after
that did the train pass the station, — sound travelling 1125 feet per
second ?
20D
ANSWERS TO THE EXAIMPLES.
1. 492480; 161280.
4. 3021; 3300.
7. 51520; 206080.
10. 996528; 73029.
13. 92160; 25200.
16. 3816; 21607.
19. 44160; 324003.
22. 1132; 37584.
25. 1096; 440.
28. 3936; 188.
1.
2. 16000; 84000.
5. 45647; 40821.
8. 6912; 394240.
11. 10708; 408584.
14. 13200; 733.
17. 126060; 15620.
20. 1180; 716.
23. 351; 361152.
26. 1088; 7040.
29. 9855; 2030400.
2.
3. 6600; 842.
6. 14161 ; 1647C0.
9. 21728; 84624.
12. 26921; 1741872
15. 4750; 16820.
18. 28624; 45780.
21. 8760; 23184.
24. 1074088; 599616
27. 1158; 1032.
30. 3960; 16815600.
1. 3751916; 3752. 2. 7329 ; 29316.
3. 1429 ; je208 6s. 8</. 4. £295 17*. U\d,i £458 7s. Sd
5. 400^. I7s. ed. ; £128 Sj, G^d. 6. £364 lis. Sd. ; 1167^. 13^. l^d
7. 16 tons 15 cwt. 1 qr. 20 lbs. ; 3 cwt. 3 qrs. 2 lbs. 9 oz. 14clrs.
8. 4 tons 1 cwt. 3 qrs. 7 lbs. 5 oz. 12 drs. ; 60 cwt. 1 qr. 16 lbs. 10 oz.
9. 2 tons 15 cwt 3 qrs. 6 lbs. ; 1 qr. 22 lbs. 1 oz. 5 drs.
10. 6 tons 8 cwt. 14 lbs. 1 OK. ; 10 cwt 3 qrs. 25 lbs. 6 oz. 15 drs.
ll.fi cwt 1 qr. 23 lbs. 7 drs. ; 28 tons 2 cwt. 2 qrs. 1 oz.
12. 6 tons 12 cwt 1 qr. 1 lb. 15 oz. ; 12 cwt 3 qrs. 22 lbs. 5 oz. 3 ars,
13. 2 lbs. 3 oz. 8 dwts. 20 grs. ; 125 lbs. 3 oz. 6 dwts.
14. 2 lbs. 11 oz. 11 dwts. 9 grs. ; 2 lbs. 1 oz. 13 dwts. 15 grs.
15. 18 lbs. 11 oz. 10 grs. ; 32 lbs. 9 oz. 18 dwts. 9 grs.
16. 47 lbs. 4 oz. 7 dwts. 13 grs. ; 22 lbs. 1 oz. 3 dwts.
17. 6 m. 6 fur. 150 yds. ; 43 lea. 2 m. 2 fur. 31 yds.
18. 15 fur. 66 yds. 1 ft 7 in. ; 71 m. 4 fur. 205 yds.
19. 8 m. 1 fur. 86 yds. 4 in. ; 11 lea. 1 m. 6 fur. 110 yds.
20. 849 yds. 3 na. ; 9098 cUs 2 qrs. 2 na.
21. 758a. In. 1 p. ; 25 sq. yds. 6 ft 69 in.
22. 125 a.; ISsq.yds. 3 ft 128 in.
23. 4 cub. yds. 7 ft. 1280 in. ; 2 cub. yds. 26 ft. 5? in.
24. 2 cub. yds. 7 ft; 1513 in. 5 3 cub. yds. ^3 ^l. IWim.
210 ANSWERS TO THE EXAMPLES.
25. 2273 gals. 3 qts. 1 pt. ; 9G8 gals. 1 pt. 3 gills.
26. 22 Ids. 2 qrs. 1 bus. 1 pk. 1 gal. ; 178 qrs. 3 bus. 1 pk. 1 gaL 2 ^ts.
27. 561 Ids. 1 bus. 1 pk. ; 22 Ids. 7 bus. I pk. 2 qts. 1 pt,
28. 278 Ids. 1 qr. 2 bus. 3 pks. 3 qts. ; 9354 qrs. 7 bus.
29. 377 yrs. 214 days ; 5 w. 6 d. 5 hrs, 23 m. 49 s.
30. 1404 w. 3 d. 23 h. } 2 yrs. 101 d. 20 h. 25 m.
3.
£ s. it. £ s. d. £ i. d. £ 8. a,
1. 12 8 1 2. 140 18 .10 3. 207 12 7f 4. 162 14 ll
5. 120 18 6. 87 1 7. J14 12 lOj 8. 169 19 Oi
9. 110 17 55 10. 82 1 10 11. 172 2 Ij 12. 193 2 2^
lbs. oz. dr. qrs. lbs. oz. cwt. qrs. lbs. qrs. lbs. oz
13. 47 1 11 14. 8 18 12 15. 61 3 16. 80 15
qr. lb. oz. dr. cwt. qr. lb. oz» tons cwt. qr. lb.
17. 12 11 5 9 18. 120 2 2 19. 43 9 2 17
oz. dwt. gr. lb. oz. dwt. oz. dwt gr, lb. oz. dwt.
20. 31 1 14 21. 84 7 9 22. 34 15 11 23. 133 5 10
lb. oz. dwt. gr. lb. oz. dwt. gr. lb. or. dwt gr.
24. 116 6 2 23 25. 107 1 10 17 26. 73 2 1
dr. 8cr. gr. oz dr. scr. dr. scr. gr. oz. dr. scr.
27. 22 2 16 28. 36 4 2 29. 37 7 30. 39 6 1
yds. ft in. fur. po. yds. m, fur. yds. lea. m. fur.
31. 58 8 32. 24 34 4 33. 21 54 34. 27 6
fur. po. yds. po. yd. ft. y«ls. ft. in. po. yds, ft. in.
35. 22 10 4i 36. 102 1 37. 30 1 2 38. 28 4 2 11
po. yds. ft. in. m. fur. po. yds. m. fur. yds. ft.
39. 32 4 7 40. 119 2 27 2 41. 27 133 2
vd> qis. na. yds. qrs. na. . ells qrs. na. ells qrs. na.
42. 167 1 43. 984 44. 328 3 1 45. 142 I
s.yds. s.ft. s.in. r. p. s.yds. a. R. P. a. r. p.
46. 1.'.5 3 44 47. 30 9 18 48. 131 21 49. 162 2 23
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53. 92 9 429 54. 106 10 8 65. 95 11 108
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56. 150 3 1 57. 103 3 1 58. 21 1 1 59. 115 1 1
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60. 119 2 2 61. 119 4 4 62. 124 5 1 63. 168 3 1
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64. 93 1 3 65. 155 3 1 2 66. 150 3 1
d. h. m. s. mo. vr, d. h. d. h. m. 8.
67. 22 2 28 59 68. 115 1 1 14 69. 20 21 49 48
7' d. h. m. y. w. d. h. y, d. h. m.
ra 32 114 21 3 71. 94 41 6 11 72. 28 184 4
ANSWERS TO THE EXAMPLES. 211
£ 8. d. £ s. d. £ s. d. £ s. d,
1. 10 3 3 2. 33 7 21 3. 60 12 2^ 4. 15 3 10
5. 55 9 10 6. 8 7 6 7. 2 18 1| 8. 187 1 2j
9. 25 17 2| 10. 38 2 Oj 11. 77 15 If 12. 215 2 31
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17. 8 U 4 IS. 1 G 2 19. 14 27 12 20. 3 21 6
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21. 3 4 10 22. 13 17 23 23. 6 7 17 24. 8 1 2
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25. 21 4 8 26. 36 8 11 27. 8 10 15 28. 14 6 6
dr. scr. gr. oz. dr. scr. lbs. oz. dr. dr. scr. gr.
29. 3 19 30. 2 2 1 31. 17 7 7 32. 1 16
yd. ft. in. po. yds. ft. fur. po. yds. m. fur. yds.
33. 1 1 9 34. 9 3 2 35. 5 21 3 36. 4 6 124
m. fur. po. fur. po. yds. lea. m. fur. fur. po. yds.
87. 12 2 29 38. 1 18 5 39. 18 2 6 40. 27 4
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41. 7 4 1 42. 7 5 43. 4 3 1 44. 4 4 2
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45. 6 2 86 46. 8 22 6 47. 6 27 48. 13 2 34
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57. 2 2 1 58. 5 1 1 59. 3 11 60. 18 2 1
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61. 5 3 3 62. 12 4 6 63. 17 1 1 64. 2 1 4
hrs. m. s. d. hrs. m. w. d. lirs. mo. w. d.
C5. 13 57 49 66. 7 19 45 67. 5 13 68. 3 2 €
yrs. d. hrs. yrs. w. d. yrs. w. d. yrs. d. hrs.
69. 12 196 9 70. 8 39 5 71. 10 43 4 72. 6 346 14
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212 ANoWEKS TO THE EXAMPLES.
£
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11. 6 ewt. 1 qr. 26 lb. 15 oz. 8 dr. 12. 41 tons 18 cwt. 1 qr. 18 lb. lOoz.
13. 159 tons 1 cwt. 10 lb. 13 oz. 14. 314 tons 10 lbs.
15. 31 tons 19 cwt. 1 qr. 6lb. 11 oz. 16. 811 tons 15 cwt. 3 qrs. 3lb. 4oz.
12 dr. 9 dr.
17. 182 lb. 10 oz. 1 dwt 13 gr. 18. 131 lb. 2 oz. 15 dwt. 20 gr.
19. 12 lea. 1 m. 4 fur. 16 yds. 8 in. 20. 19 lea. 2 m. 1 fur. 98 yds. 8 in.
21. 414 A. IR. lOP.
22.
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23. 319 sq. yds.
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24.
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ANSWERS TO THE EXAMPLES. 213
11.
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1. 2S 17 Hi 2. 17 Oi 3. 1 7 2 4. 19 16 6|
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5. 147 & 2s, 6d. ; 1090 & 4d. 6. 3150 ; 285 & 5s.
7. 138 lb. 6 oz. 10 dwt ; 6 dr. 1 scr. 4 gr.
8. 24 lb. 3 oz. 13 dwt. 8 gr. ; 12 dwt. 12 gr.
9. 597 & 2 qr. ; 4 & 8 in. 10. lOOU ; 550.
1ft.
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214
ANSWERS TO THE EXAMPLES.
17.
1. 21 lbs. 4 oz. 16 dwt. 10 gps..
3. £94 195. 2i.
6. 24857 mi. 1680 yds.
7. 1907314.
9. 6 da. 22 hrs. 40 min.
11. 132 yds. 2 ft. 7 in.
13. 976 ducats.
15. 365 da. 5 hrs. 48 min. 48 sec.
17. 44 tons 12 cwt. 3 qrs. 12 lbs.
19. 1 mi. 3 fur. 216 yds. 2ft.
21. 3 mi. 3 fur. 60 yds.
23. £7670.
25. 63 yds.
27. £193 155.; 60min8e.
2. 29 da. 12 hrs. 44 min. 8 sec.
4. £16 45. lid.
6. £21 195. 6d,
8. 250 ft
10. £91 105. ed.
12. £387 15. 1^,
14. £19 45. 0^,
16. 15 cwt. 7 lbs. 8 <w»
18. £9895 165. Sd,
20. 91717720 mi.
22. £13069 05. 7d.
24. 105. 4^,
26. 13.
28. 114 lbs. 15 dwt. ; £3437500.
30. 12 da. 2 hrs. 24 min.
29. £1919 55. 5d,
31. £12389 l5.3i.; £10110 185. 9f?. 32. 20833^ lbs.
33. 37 oz. 34. 648.
36. 50606 gal. 36. £664.
37. 26 yds. 2 ft. 38. 355 sq. yds. 7 ft. 126 in.
39. £33 25. e\d, 40. £22 75. ed,
41. 102700 cub. yds. 16 ft. 1152 in. 42. ^d.
43. 175. 4d. 44. 168 tons 7 J cwt.
45. £148 105. 46. l5. life?.
47. 1607 tons 2 cwt. 3 qrs. 12 lbs. 48. 7 mi. 2 fur. 120 yds.
49. 235. 4i<f. 60. £4 145. 7Jc^. ; £5 85. 2d.
61. 2l5. 62. 65. 2d,
63. 13 ac. 2957 sq. yds. 7 ft. ; 10 ac. 1477 sq. yds. 7 ft
64. 353571 tons 8 cwt. 2 qrs. 8 lbs. 55. 68| yds.
66. 725 gal. 67. 6044.
68. A man, £16 105. ; a woman, £5 105.
69. 7. 60. 3 ac. 684 sq. yds. ; 10 ac
61. 20. 62. 750 bu.
63. 2 yrs. 334 da. 19 hrs. 30 min. 64. £9 35. U. ; £5 85. 4d ; £i: 85. id.
65. A man, £66 05. i^d. ; a woman, £33 05. 2ld. ; a child, £li (te. Ojd
66. A, 75. 3^6?.; B, Us. ll^d.; C, 27s. lid. ' 67. 10240.
68. Gross totals, (i) 73172; (ii) 84822.
ANSWERS TO THE EXAMPLES.
215
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216 AKSWERS TO THE EXAMPLES.
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ANSWERS TO THE EXAMPLES.
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218 ANSWERS TO THE EXAMPLES.
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7. i. 8. ^. 9. 26fft. 10. 3^.
11. 9 oz. 3 dwt. 8 gr. ; 145. 3d 12. 3^ ; lOSj sq. in.
13. £5 6s.Sd. 14. 3l;lJ-|. 15. |.
16. -jI^. 17. 1. 18. J. 19. fig.
20. 5/g;2|g. 21. £67 4s. 31(1 22. l^j |1; 2^
23. £85 145. 3f</. ; 45. 7d, 24. 4|; 42. 25. £21 85. l^d.
26. £9973 65. Sd. 27. ^^llL^f. 28. 4M. 29. £90.
20
'64*
30. 19 dwt. 9 gr. 31. 14^;^. 32. 240^^0jJ03
o4
33. IJ; £7 165. ali. 34. 2l5. 35. 23 lbs. 17 dwt 5jgr.
36. Iff. 37. £4 165. 38. £l IS*. 7|rf.
39. 59 7ds.; £11 l5. 3cf. 40. ^;£3125. 41. ^; 680f lbs.
42. f. 43. 99. 44. 9.
45. MJa'i^ill; sum, Iff. 46. 81.
#/. 12^. 48. 140|yds.; £1 65. 3|i.
49, 17 ewt 2 qw. 5 lbs. ; £32 168. 4d, 50. £333 6$. Sd.; ^.
ANSWERS TO THE EXAMPLES.
219
37.
].
3.
7.
0.
11.
12.
•7, 11-7, -33, 1015.
•230037. 4. l-lllll.
37 _JL__ 1 3
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11 3 9,Q_61_
16000' 32U0> ^"1600*
2. -01, -0021, -0117, -0000003.
5. 13-003005. 6. 10 110101.
Q _3 19 3 O 9
"• 400' -^40' 16' "40*
10 15i^ -^— 4-1-
iv/. At'oi' 12H0» *128*
3, 300 ; -03, -0003.
•125, 12-5; -0000125, '000000125; 5387340, -0538734.
1100, 1100000; -0011, '0000011; 11025, 1102500;
•00011025; 213012000; -000213012.
-011025,
38.
1. 34-62156. 2. 7828594. 3. 420-615973.
4. 2492 2622123. 5. 19002 : 3-44902. 6. 2M335 : -41213.
7. 19-0002 : 1-0013. 8. -0000013 : 23016484.
9. 1-33678 : 2-7486. 10. -003213 : -34235.
39.
1. 723-6 : H6 4561.
3. -07504 : -000602.
5. 6-31441 : 4-096.
2. '0000001 : 74-151.
4. -0013014 : 1-5.
6. -0001234321 : -00044408.
1. 6-25 : -000625. 2. 6250000 : -0000625. 3. 490000 : 6 3.
4. 185 : 30. 6. 4000 : 4-8828125. 6. 24 : 1200.
7. -00015625 : 7118580. 8. -0122699 &c. : 1568627 &c.
9. -3388278 &c. : '00383177 &c. 10. 290 : '014974 &c.
ftl.
1. -04 : -052 : 5 25 : l-6. 2. -848 : 11-0136 : 15-625 : 61875.
3. 7-203125 : '1328125 : '00015625 : 11-001696.
4. -001953125 : 10009765625; -008125; -0013671875.
5. -1705; -00216; '32.
1. 1-4 : '572 : 2-345 : '01256. 2. 2-9^85714 : -5 045 ; -0132 ; 23156.
3. -6089 : 6-761904 : 17-12951 : -12345.
4. -03646 : -1003376 : -40664 : -020502.
6. -0588235294117647.
•0434782608696652173916.
-0344827586206896561724137931.
•03226806451^12^.
220
ANSWERS TO THE EXAMPLES.
].
3.
6,
1 . 5 . 6 . 27
3 • 99 • 11 • 37*
o23 • _13^
"SS • 3000
9 101
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202
1-8. •
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6 2-?-* fiil- 1115
1. 47-411455286.
4. -657142 : -0058.
7. 3-6 : -062.
2. 168-7023011466.
6. 9 928 : 2-^97.
8. 49 : 1-145.
3. -24 : -0327116.
6. 31-791 : 352 08564.
ftS.
1. 9s. : 13s. 7 id. : £2 68. 6d, 2. £8 2s. 6d. I 6s. 2i. : £l lis. 8i.
3. 13s. l^d. : Is. 616^. 4. £18 2s. Sd, I 9 cwt. 3 qrs.
5. 23 d. 10 h. 4 m. 48 sec. : 1 A. 1 b. 35 p.
6. £1 14s. 3ci. : £47 5s. 7^ 7. £8 9s. Z^d. : £125 13s. lOje?.
8. £1 lis. Gld. : 10s. \id,
9. 13 p. 2 yds. 1 ft. 4 in. : 21 lbs. 12 oz. 7*68 drs.
10. 3 sq. ft. 67i in. : 102 m. 875 yds. 5*76 in.
11. £78 3s. l-8645<i. : £120 5s. 9-3125«^.
12. £2 Is. 3-50625<?. : 6s. 6d. 13. £l 3s. O^d,
14. 12s. IK 15. 10s. Ud. 16. 15s. id. : 17s. 3|(f.
17. 85 m. 7 p. IJ yd. : 73 A. 2 p. 20iyd«!. 18. £7 13s. ^d. I Us. 3^.
19. 75. lljfi. : 8s. 7H 20. 16 lbs. : 1 qr. 4 lbs.
1. -475: 021875. 2. •375:1-725.
4. -125; 27 5. 5. '3125; •196875.
7. '875; -5384375. 8.
9. -19453125; -03625. 10.
11. 2-6; 1-424. 12. -00022095; -924.
14. 97-6; -377083.
16. -127109375; 6156510416.
3. 1125 : -2625.
6. -5703125; -39375.
•777587890625; -05.
-039375; "046875.
13. 1-86; -869375.
15. 4-90; 4-2083.
ft7.
1. If. 2. ^d. 3. 395S miles nearly.
5. -02734375, 36-671428; 3}, 3f|, -0004935, '282.
6. -375; £2 13s. Sd.; 7^
A 'J3$, 4'2li2857; ^H^; 530, -00341.
4. 3J days.
7, 16s. llfei.
9. 10s. Sid,
10. '33ri42$; 8-75.
11, 7 K U m,-, \ k. ^ u. 13 p. 22 yds.
a:?svvers to the examples. 221
12. ll|lf= 11-8208. 13. £dis,Sld,
U. 45. dd, = l'd of 2s. 6rf. 15. £i6d 16s. l^d.; £127 Cs. Cd.
16. £2 lis. id. 17. lis. 3<i.
18. -06040625, 'OOQd; fo'u. nfe; ^3 13s. l^d, 19. 3s. Hit?.
20. 2i| = 2-59375. 21. £Z 2s. Uf^d.
22. 16 ft. 104|§ in.; 20 ft. liSS^^ in.
23. -18988; 025; £4 4s. iid. 24. £25 17s. 2|-ld; 7s. 2Jei.
25. £4 4s. 9irf. 26. 76iyds.
27. £2 Os. 3ld.; £6 6s. 6f(?.; 4-78126.
28. 8176; 816; 27; '75; 135-1940625.
29. Is. 9|c?. 30. -08286.
31. -109375, -1076923; i§,g*^; -54140625. 32. £15 14s. lOf^i.
33. 2-625, -036, 2^, fj^; 3971875.
34. 7 cwt. 3 qrs. 8| lbs. ; £8 13s. 7d, 35. £81.
36. 2-140625. 37. '03. 38. 1-1457; 423; 18s. 6|<?.
39. 69-0625. 40. £410 lis. 9-J§2<?.; £41 lis. 10.^<^,
41. £32 15s.; 4192; 1250.
42. •021484375,-06; 2^^^, ff^; -C 009765625.
43. -0875; 4-67. 44. £14 Os. IJd^.; -4. 45. 9.
46. £2 4s. lid.; 24 p. 5025 eq. yds. 47. £34 14s. SUd.
45. 3-14159. 49. £3 6s. e^d.; 6-66625. 60. 2-7182818.
£ s. d. £ s. d. £ 8. d. £ s. d,
1. 838 10 2. 1486 6 8 3. 1452 6 4. 2606 8
5. 2213 3 4 6. 210 7 6 7. 3203 18 8. 6212 5
9. 819 10 10. 2500 11. 3459 11 8 12. 1777 2 6
1. 476 6 2. 1263 2 3. 1559 8 4. 2344 7 6
6. 1986 1 6. 879 14 6 7. 4279 12 6 8. 4455 1 6
9. 377 12 6 10. 3374 11 6 11. 3413 5 12, 645 14 9
50.
1. 23 17 7| 2. 24 1 If 3. 179 13 11 4. 86 13 3
5. 103 8| 6. 143 15 2^ 7. 79 6 7J 8. 361 16 8
9. 286 15 6f 10. 129 17 dl 11. 284 6 4^ 12. 448 11 7i
222
ANSWERS TO THE EXAMPLES
£ a. d.
1.
400 4 4.i
4.
24S6 15 7
7. 2542 01
10. 366 13 2|
51.
£ s. d.
2. 1059 9
5.
8. 2696 5 10
11. 1841 7 9J
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3. 1070 2 Of
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9. 201 14 9i
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4.
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6. 51 16 6^
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15. 105 17 52J
16.
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19.
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2.
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3. £2610165.111^.
4.
£4713 Is. G^d.
5.
£10369 Os. lOi.
6. £48 65. IQld,
7.
£22 19s. n^l.
8.
£1144 05. ll|g<?.
9. 313a. Ik. 18p.
10.
£473 lis. Qi^.
11.
£1912 14s.
12.
26 11)8.2 oz. 11 dwts. 16 1
»rs.
13. £223 lis. 3^.
14.
£80 175. 2ld.
15.
£26 6s. 6d,
16.
£155 9s. 2c?. ; £2 13s. dd.
17. £273 65. 6d.
18.
£1191 10s. lid.
19.
£55042 Is.
20. £6 10s. 9^.
21.
£173 9s. ^d.
22.
£2560 14s. S^d.
23. £25 Os. Oif^;.
24.
£2430 85. 1^.
25.
£7 9s. 7id.
5ft.
1.
11 92 02 a
'2» ^3» ^3» "•
2.
2|, 3|, 35, 6§.
3. 31,4^,41,71
4.
4f, 5g, 5g, 8f.
5.
If, 21 2i, 171.
6. 2l6f, 6f, 10.
7.
If, 2f, 2f, 311
8. 5, 5, 5, Of.
55.
1.
£10. 2.
207.
3. £72.
4. 30.
5.
35. 6.
£55.
7. 210.
8. 378 yds.
9.
£50. 10.
£10.
11. 39fqu.
12. £4 65. 3e^
ANSWERS TO THE EXAMPLES.
223
1. £6% Zs. 2d.
6, 75a.. 2b. IOp.
9. Is, IgOO^'
5«.
2. £6 108, 2^. 3. 176 m. 4. 1 h. U m.
7. £1 Os. llld. 8. £4 158. 6^-.
10. £11 95. 4^.
6. 13^. 3<^.
57.
1. 150.
5. 4.
2. 6 mo.
6. 8^<?.
3. 12 mo.
7.
622|A.
4. 171.
8. 81 oz.
58.
1. U. n^i, 2. £37 12«. 6d.
6. £19 125. 6. 165 cwt. 19}? lbs.
9. 2 cwt. 2 qrs. 15 lbs. 5 oz.
11. £2094155 165. lO^d,
4. 135^ bu.
13. £7144 75. 6d,
16. £11 ll5. 2ifc?.
19. £1451 175. O^d.
22. 178 ft. 11^ in;
26. 286^ m.
29. £33 185. 4d,
32. 10s. e^,
35. £270.
38. £5 175. n^.
24.
14. £26 185. 2Hd.
17. £3.
20. £450.
23. 6|hrs.
27. £79 105.
30. £1 165. 9d.
33. Il5. 4Jc?.
36. 7722.
39. £13 95. (^.
3. 55.
7. 35. ed. 8. 170.
10. 2 lbs. lOfoz.
12. £79 l5. 7^d.
15. 640^ yds.
18. 65. 3||<?.
21. 85 days.
12800. 25. 72.
28. £8 35. 8|fc?.
31. 85. 5^.
34. 4|yds.
37. 32 ft. ; 152 ft
40. 26^ lbs.
59.
2. 27. 3. 16. 4. 15. 6. 12.
7. 125 rms. 8. £194 85. 0. 14 wks. 2 da.
11. 45. 12. 112. 13. £520
15. 615. lOje?. 16. 6f da. 17. £545 65. dd.
19. 34 mi. 20. 8. 21. 240J.
24. 6 tons 17 cwt. 16 lbs.
1. 44 da.
6. 3 12 J lbs.
10. £114 65.
23. 2^ days.
26. 10| hrs.
14. 9.
18. 3 wks. 6 da.
22. 13 J.
25. 4.
27. 182. 28. 8. 29. 121 days. 30. 50.
1. £125. 2.
6. £247 165. 7^.
8. £71 125. 2f^.
60.
£45. 3. £1260.
6. £2857 105.
9. £37 175. 3^.
4. £2673 25. ed.
7. £744 16*. lid.
10. £20 105.
Q2
224
ANSWERS TO THE EXAMPLES.
^ «. <^
1. 519 19 lf§
4. 19 10 11^
61.
£ 8. d.
2. 7612 7 5^
5. 492 4|
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3. 1196 19 6
6. 284 6 1|
62.
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2. 57 17 7
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5. 26 5 5
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63.
1.
5§. 2.
£42 5s. lOd. 3.
125 days.
4.
6.
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25 yrs. 6.
£39
7s. 6a. 7.
2i.
8.
£1043 15«.
9.
3|yrs. 10.
2|.
11.
£8 8«. 2|d
12.
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6ft.
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2.
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3. 199 1 3
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6.
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8. 17 6
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4.
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5. £90.
6.
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7.
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8. £771 7s. 6d.; £lO 12«.
6d.
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10. £25.
11.
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12.
Increase of £20.
13. £16 I3s, 4d. 14.
£53 65. Sd.
15.
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16. The 3i per cents. 17.
£1
7 Ss. 6d.
18.
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£4241 17 s. ed.
19. £715.
20. 93|.
1.
4.
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1U;£210 8«. 7i^. 8.
£93 65. Sd. ; 11 J per cent, 11.
£44 I5s. 14. 40. 15.
£4 9s. 7gd. 18. 5| p. c. gain.
8 p. c. gain.
9s. 2irf.
19.
3. £2 45. 6^.
6. £30 165. ; 9|.
9. £1 05. 2ii.
12. £82 105.
16. 2| p. c. loss.
25. 20. 63jt
ANSWERS TO TlIE EXAMPLES.
2211
68.
1. 213, 365, 497 ; 626, 315, 226. 2. £72, £09, £108.
3. C, 16 cwt. qrs. 20 lbs. ; T. 1 cwt. 2 qrs. 19 lis.
4. £46 IZs, 4d., £35, £28, £23 6s. 8i., £20. 6. 14, 112, 378, 896.
6. 0. 889 oz. ; H. Ill oz. 7. £66 Us. id. ; £33 6s. Sd. ; £200.
8. £6 17*. Zd. ; £4 165. Z^. 9. 3 oz. 7 dwt. 6ft grs.
10. N. 1702f lbs. ; S. 212J lbs. ; C. 324J lbs.
11. 1 lb. 11 oz. 10 dwt. 20i^ grs. 12.
13. £160, £175. 14.
15. £28 2s. 6d. ; £86 35. l|rf. ; £11 Us. i^d.
16. 12 carats. 17. 15 carats: 15 oz. 18.
2 oz. 4 dwt. 14 grs.
£102; £104; £78.
19. 4«. 2^. ; 6s. 7^. ; Is. dd.
15 carats.
20. £100; £300.
69.
1.
5.
2.
358|.
3.
3. lbs. 2 oz.
4.
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6.
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6.
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7.
£96-10561.
8.
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0.
mi
10.
86*186 : 1,
70.
1.
a.
6.
7.
9.
10.
11.
2. 185; 371.
4. 309; 499.
6. 6123; 4117.
8. 9998; 4908.
73; 94.
729; 592.
690; 80700.
1880; 8097.
346761; 607002.
2-828427+; 4-472136- ; 19052559-.
187-403308+ ; 94005319-. 12. 3673; 806-54.
13. 81-6279-; 6-270009 + . 14.
15. '1096- ; -0398. 16.
fl; -6060915+; -66789 + . 18.
16-9595+; 78i; 191647-.
•0674485- ; -096386+; 1-426353-. 21. 4|; 1-103026.
925 links. 23. 38 ft. 9 in.; 54 ft. 9*6 in. w^aWy. 24. 1045 yds.
17.
19.
20.
22.
•47434+; 7-1505.
-009076; -144914-.
•3118048- ; -2400274+; l\l
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71.
1.
67; 74.
2.
28; 190.
3.
163; 328.
4.
456; 9870.
6.
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6.
6397; 5608.
7.
7099; -369.
8.
36-8403+; 8081 +.
^.
20-03909+; 17-84109 + *
10.
•941036+ ; 3-1 158 + .
11.
1 ft. 10-56 -in.
12.
852296 miles.
226 AKSW£BS TO THE EXAMPLES.
MISCELLANEOUS. 72.
1. 18880. 2. £345. 3. f ; Iff ; Is. l^d.; 3^.
4. £1 17s. 7id. 6. £1492 13s. 7^. 6. ISIJ qu.
7. 41J ft. 8. £318 16s. 9. 31s. 6d.
10. 2a. 1b. 6|fp. 11. 609; 85 ft. 10 in. 12. 2676f|| qrs.
13. 3|. 14. 20. 15. 12g%. 16. Is. 2|f<i.
17. 4s. Sd,; ij; f^. 18. £3 2s. Ofjci. 19. 80976; 4578082-.
20. £150, £180, £240, £300. 21. 6|| days. 22. 96^.
23. fir, 162^^0; l^; ^^J 2308. 24. A, 80s. Sd,; B, 778, 9d.
25. £11 6s., £20, £29 6s. 26. £70 lis. 9^. 27. 23i days.
28. 610 9-. 29. £6 6s.; £4 3s. 4<i.; £3 2s. ed.; £2 10s.
30. 884, 163. 31. 3 h. 20 min. 32. |f. 33. 105 da.
34. 3035913 +; 960040103 + . 35. 5|| da., or 6 da. 7^ hrs.
36. -68126; |; -00266266, 266'256, -0266256.
37. 4s. lOld, 38. 59 min. 8i||§| sec.
39. 13s. 2§f^., 6s. 7^., 3s. Z^, 40. ^. 41. £14 13s. eld.
42. £1 19s. 6^^. 43. £3200, £4800, £6000, £7000.
44. 3 tons 17 cwt. 2 qrs. 26| lbs. 46. 45. 46. £11 19s. i^d.
47. -25298 &c.; 5|. 48. £4 7s. S^.
49. 69^ degrees =:76i§ grades. 60. Uff hrs. 51. ISfJ.
62. f of a da. 63. 21s. Sd, 64. £l8 10s. 4^. 66. 1500,
66. -36; 25 lbs. 15 oz. 11 904 drs.; 4 miles 30| yds.
67. £759 6s. 7^Sid. 68. 625; 12-84. 69. £3499; £874 15s.
60. lOifp. 61. 111104. 62. £3 7s. 2d. 63. -057 &c.
64. £34 10s. lOlfi. 65. 3||; {% 66. 5^5 in.
67. 10s. 5d. 68. 245 : 243. 69. 25-416 ft.; 11-747 ft.
70. 2s. 7id. 71. £49 9s. 4i. 72. ^; -315625; £2000.
73. £49. 74. 6315 dollars 55^ cents.
75. £33 6s. 3|g(?., £66 12s. 7^., £99 18s. UfJ^i., £133 6s. 3^cf.
76. 42 m.; lOJ m. ' 77. 6 per cent. 78. lOd,, Is. 4Jd, Is. nd.,&c.
79. 678||. 80. 105. 81. Is. 9d., Is. 2d., and 7d,
82. 3s. 5^. 83. 560-22 &c. 84. IJ min.; 427^, 190.
85. £907 10s. 86. ^f^; -69140625. 87. £6 2s. 2J<i.
88. £15. 89. 107 yds. 2 ft. 11 in.; £6 14s. 11^^.
90. £32 Is. l^.; Zd. 91. 99^; £l76 4s. 2||?i. 92. £123 lis. 4di
93. £3 4s. lid. 94. 23^"^^; £1 14s. O^d.; 8s.; -06516625.
95. 1 lb. 3 oz. 7 dwt. 412 grs.; 1555x^. 96. 6f|||.
97. £316. 98. £319 16s. 8^.
99. 790079 +; 37-9241-; -069; 30-02. 100. 8 days. 101. 16 hw,
J 02. 6 min. 17^ sec. before noon, 103. 6001f| yds.
J04. B, 6d.; C, 28. 6d, 105. £1011 Os. Z^.
ANSWERS TO THE EXAMPLES. 227
106. ;e840, £795. 107. 89iJ|. 108. £1050. 109. £10560.
110. £18668 28. 7Uld. 111. 65. 112. 135. 9d.
113. 1 hr. 61f mic. 114. 15 cwt. 115. 3 ft. 9-02221 &c. in.
116. A, £16 Is. 8d.; B, £8 6s. 117. £11 16«. U.
118. -095178+ ; 21^; §. 119. £1706 13s. U. 120. 3i.
121. 18s. b^^. 122. 9 days. 123. A, 264; B, 198; C, 308.
124. £300. 125. 2133J. 126. £2771 7s. 0^^.
127. 7 ft. 4| in. 128. 16s. M. 129. £22 13s. 2%A.
130. £1 8s. 6?i. 131. £410 lis. ^H^.- £41 lis. 10J<?.
132. £36893 6s. U, 133. £6 8s. 10|<?.
134. 3s. 4(^.; dd. 135. £3 15s. 2|<Z.
136. -45. 137. 18| per cent.; 10s. Q{^d. cost price.
138. £2027 Is. 1^^^. 139. 33V 140. 1. 141. 3s. U.
142. £8 7s. 143. -05099902-; -0155048+ ; '9615-.
1-14. 3| ft.; 8 tons 3 cwt. 3 qrs. \^ lbs. 145. 1 percent.
146. £595 Os. 9^. 147. 8 hrs. 30 min.; 10 hrs. 22J min.
148. 121 J. 149. £3 10s. 9^.; '77. 150. 63.
161. £4957 Qs. U. 162. £l 8». 9^. 153. 87^.
164. £220, £6 Is. Ud. 165. 10s. 8^.
166. 30s., 15s., 10s., 7s. 6i., 6s., 5s. 157. 415-8, 356-4, 226-8.
168.600. 159. £12800. 160.26^. 161. 4* lbs.
162. ff, -00390625, %-^, 1^. 163. 121j. 164. £130.
165. £245 18s. 11|^. 166. 16. 167. 4^, £-293 6s. U.
168. £292 4s. 169. £62 3s. ^^., 34733*92.
170. l-4d, 6-483; 2-49, 8-57. 171. 550 tons, OSf.
172. 18s. 2\d., lis. 9ld. 173. £65 15s. 9xV'
174. 6384, 7695, 8321; 2-'*^ da. 175. £5 13s. Old.
176. ^. 177. f, -9147916. 178. 12 hrs. 8 min.
179. -6. 180. 91^.
181. 273-649. 182. £520. 183. 2880, -00994316, ^ff.
184. The 3J per cents. 185. £187 10s., £312 10s., £500.
186. 4 lbs. 11 oz. 19 dwts., -165234375, f^, ^, f^.
187. 266 tons, 16^ cwt. 188. 176a. 640 sq. yds.
189. 7s. Ojfa^i. 190. £270; £11 8s. Zmd.
191. 2400, 1800, 1600, 1500.
192. £5 14s. Q'ld,, £182 10s., £6 16s. I0\d.
193. £3250, £1560, £1440. 194. 80 and 160. 195. 66-286.
196. £211 19s. Zd, 197. The 3 per cents.; 19s. 71^. 198. 8^.
199. 6|, i?, llf. 200. £532 45., £100 16s., £492.
201. -45593- ; 70-61. 202. £94 10s., £7 85. lOjV^. 203. 61, 1||.
204. £127 5s. 5^., £127 12s. \d. 205. 20-7846 &c., 203646 &c.
206, £196, £304. 207. 491xio- 208. £6 Os. 11^.
209. £2 58. 210. £320, £293 68. %d., £\\Q, «aQ\ V^s» '^^^
228
ANSWERS TO THE EXAMPLES.
211. 4^. 6|^.
213. £10 Ss, 214. 3«. 1|^
212. 687g.
215. £1832 19«. 6^rf. 216. £29 17«. 2^.
218. 1 ton 12 cwt. 2 qrs. 3 lb. 5 oz. ; £8 14^. 6|<2.
219. 12 hrs. 48 min. ; 4|, 5|. 220. 224 miles 64 yds.
221. 88 min. 222. f^, Jt, IJ, /_.
223. Loss, £7843 155. 224. £22 10s ; 2^^ per cent.
40^
217. 12 days.
EXAMINATION-PAPEKS.
Paper V.
3. 31 fiq. po. 30 yd. 2 ft.
5. 19 ac. 2 ro. 29 po. 2 yd. 5 ft. 81 in.
7. 1 ac. 2 ro. 3 po. 4 yd. 5 ft. 6 in.
9. 668 sq. yds. 10. 1224*6 gall.
12. 3'962 met 13. 160*93 decam.
4. 12524940 in.
6. 17778376 in.
8. 278971 ft.
11. 31*103 ft.
14. 100000. 15.
8^*
4. 18 : 20.
7. 96 : 80 : 120 : 105.
5.
8.
Paper VX.
8 : 13. 6. 7 : 15.
1 : 3or|. 9. ilfto^as 18 : 17.
1. 77|. 2. 24/3|.
5. S^d, 6. 20^.
9. 4/li.
•
Paper VXX.
8. £22241170. 4. 40° 63'.
7. 7 sq. ft. 8. 60.
10. 21|. 11. 3 galL
6. 22|da.
Papdr VXXX.
7. A, 33| hrs. ; B, 24 hrs. ; C, 18| hrs.
8. A1%S15, C20da.
10. 360 gall. ; 1 gall, pet hr. gained.
9. 51 da.
1. £89 4^. 4l<;.
4. £323 3«. l^.
7. 2211 dot. 16^ re.
10. 45.; 42^ftancs.
Paper XX.
2. 6576 ft. 51J cts. 3. 95286*21 £r.
5. 9*386<?. 6. 62yi. nearly,
8. 53J<?. per milree, nearly, 9. 3722*07 fr.
11. Gains 85. nearly.
12, CIivuitou$ly, by 35*986 mibces.
13. £160 145. S^,
ANSWEHS to E^tAMlNATlON.PAPfiftS. 229
14. 480 tr. 24| cents. 15. 5 doll. 59 J cents.
Jj6. 1 rupee 11.13 annas per lb. 17. (i.) '0102045 oz. ; 26'17 franca.
17. (ii.) 25 fr. 531 cts. ; 25 fr. 14j cts. 17. (iii.) a. '088 p. C. dearer.
17. (iii.) b. '367 p. c. dearer.
Paper X.
3. 32 oxen. 4. 4. 5. 0^ da. 6. 20 wks.
7. 3 ac. 8. 40 oxen. 0. 21 daya
10. 14076 min. ; 1^ of the cist
. ^«7O0
Paper acz.
4. 264 at 125. &;c. 5. 42 and 48. 6. 40 or. ; 45 lenL
7. T. 3«. 9<f., C. l8. Sd, 8. 6 : 4.
Paper XZZ.
7. -2031. 8. 21 po. 2J yd. 9. 6 po. 1 yd.
10. 108-097 yd.; 305|yd. 12. 140. 13. 14*02 ft.
14. 153 mi. 15. 4^19; a/3. 16. .0261 in.
17. 250. 18. Sid. 19. 12 ft. 20. 433 nearli/.
21. 2 ft. 2 in. nearly; 28| sq. ft. 22. £42. 23. 4.
24. 6-5413 ft. ; 5-058 ft. 25. -.
31
26. 13-6801 cub. yds. 27. 551 p. c. nearly.
Paper XZZZ.
1. 57 min. 2. 263 times ; *0029 rem. 3. 102.
7. 46 sq. ft. 0' 0" 11'". 8. 287 sq. ft. 2' 6" 6'".
9. 46 sq. ft. Oil in. ; 287 sq. ft. 29§ in. 10. £10 Is. 9^.
11. £6 48. 6d., £Z Ids, 7d. 12. Gain 25 p. c. 13. i5^ ac
^ 2002
li.2s.7]d» 15. -7^. 17. ^ per cent, gained.
6
18. lOd. 19. Nothing, 20. 8283.
21. 3 yrs. 100 da. 22. 6| mths. 23. Value = 242 J da.
24. 18, 27, 24, 30. 25. A 5«., B 1$. 10j<f., C Is, l^d,
26. i^. 27. A 11». 4i., S U. 4d., C 7s, id.
315
58. £16 13«. id, 29. Nearly £3 I65. lid. p. c.
SO. 40 ac. 9 po. 10 3*ds. 32f in. 81. £12 Gs, 11}<^.
82. £14 10^. lOYld, 83. £260. 84. 4|1.
230
ANSWERS TO EXAMINATION-PAPERS.
Paper XXV.
2. 192Jft.
1. 3769.
4. 1 gaU. water to 17 spirits.
6. £821 55. ; 32 days.
8. 12 weeks.
10. '00416 and '0625.
13. 301|c. yds.; 166*19 lbs.
17. £1606.
19. IfJ yr., or 1 yr. 164 da.
21. £24 increase.
23. ?.
3. 391 J rev. ; 7J and 13| ft. circumfi
6. 1620 tons.
7. 1 mile, 1557| yds. nearly,
9. 729, 432, 3348, 27.
11. 165. 4<?. 12. Z\8,A\d,
16. 4 florins. 16. 31^? and 609^.
18. £127 Is. 6d,
20. 12 yards from ^.
22. £10 165. decrease.
24. '008.
20. llfmths.
Paper XV.
11. 12 men. 12. 2 pon. 13. £6947 185. 4d,
15. 3-627 p. c. 16. 7 men. 17. 145. 3^d,
19. 265. Bd,y 335. 4<f.
21. At 24 min. and at 30^ min. past 11.
23. 22 yrs. ago ; 18 yrs. hence. 24. 4.
25. 9^ mi. an hour.
27. 9^ min. past 8.
29. £1 11 5. Zd.
31. The whole.
33. £1400.
35. 3-4408. 36. 7§ mths.
38. 6 mths. 39. £147.
14. 3570.
18. Q^.
22. £104.
26. 685.
28. £126000.
30. 92 days.
32. £1078 ll5. 7d. nearly,
34. 6J p. c. ; £674 135.
37. 3 : 7.
40. £600000.
41. £1000. 42. ^.4^.; 3l5. Zd, 43. 14 cwt. 3 qr. 13 lb., &c.
44. £17 165. 4^e?., £8 185. 2^d. &c.
46. £322, £627 45., £2060 165. 46. 23li p. c,
47. For ploughing the field with oxen, £4 75. Qd, ; For ploughing it
with horses, £3 185. 9|£f.
48. 71 hours. 49. 10| hrs. 60. 6 min. 6.08 sec.
281
ARITHMETIC PAPER SET TO SENIOR STUDENTS AT THE
CAMBRIDGE LOCAL EXAMINATIONS,
In Decembbb 1880,
"with specimen "workinas fob the gthdancb of candidates.
1. Find the quotient and remainder arising from the division of
359907 by 789.
If in dividing a number by 336, the operation be performed by short
division by employing the factors 6, 7, 8 in succession, and the several
remainders be 1, 2, 3, what is the complete remainder ?
2. If a person's income for the year 1880 be £37 15^. per calendar
month, and he spend at the rate of £l 48. 7^. per day, how much will
he have left at the end of the year?
3. If 44 yards 1 ft. 9 mches of cloth cost £42 Is, 9d., what will be
the cost of 74 yds. 11 inches?
4. Simplify :
237655
(1)
367285
2 + 5+T+14+28
Reduce 22 days 4 hrs. 35 m. 42 sec. to the fraction of 34 days 20 hrs.
56 m. 6 sec.
5. Divide :
(i) 121'9192 by 3-04.
(ii) 12-19192 by 30-4.
Find the value of '07890625 of a ton, and simplify
1| of Ifl of -M.
' -61 '416
6. Find by Practice the cost of
(i) 4112 things at £l Is. 7id» each.
(ii) 3 acres 1 rood 3 poles 19^ tK^. yde. &\> £\\^ "^t ^^^^
232 EXAMlNATION-FAPEfi.
7. The breadth of a room is half as much again as its height ; its
length is twice its height : it costs £5 58, to paint its walls at l^d, per
square foot : what are its dimensions ?
8. Three men can do as much work as 5 boys : the wages of 3 boys
are equal to those of 2 men. A work on which 40 boys and 15 men
are employed takes 8 weeks and costs £350 : how long would it take if
20 boys and 20 men were employed, and how much would it cost ?
9. A walks to a place at the rate of 4| miles per hour : at 8 miles
from his destination he meets J?, and turns back with him (walking at
B'b rate) for a mile : if ^ is half an hour late at his destinatioo, what
is ^s rate, and at what rate should A have walked after parting with
J? so as to arrive at the proper time ?
10. If the difference between the interest and discount on a sum of
money for 2 months at 4^ per cent, be 2s. 3d. find the sum.
11. If the price of the 3 per cent, stock be 96, a person Can obtain
an annual income of £l more than he can if the price be 97. How
much has he to invest ?
12. In an election A receives promises from half the constituency,
£ from }§ : 400 voters however had promised both candidates, of whom
300 Tote eventually for B and 100 for A. The electors who had not
promised either do not vote. B wins by 80 votes. What were the
numbers for each?
ANSWERS TO EXAMINATION-PAPEE. 233
EXAMINATION-PAPER.
Answers worked out
1. (a) 789)359007(456 the quotient, 123 the remainder. Ans.
3156
4430
3946 .
4867
4734
123
1 . (b) Complete second remainder =6x2 + 1 = 13;
Complete final remainder =6x7x3 + 13 = 1 39. A7is.
2. Income, £37 Ids. a month, for 12 months, = £453.
No. of days in 1880, a leap year, = 366;
Outlay for 366 days= £1 4s. 7 id. x (6 x 12 x 5 + 0).
6^
7 7 9 = 6 times.
12
88 13
6
443 5 « 360 times
£453 -£450 12 9 = £2 7s. Zd. Ana,
3. 133 ft. 9 in. : 222 ffc. 11 in. :: £42 7s. 9(Z.
12 12 5
1C05 is to 2675 3 )211 18 9
or 321 is to 635 £70 12s. Ud, Ans.
or 3 is to 6
(l\ 237665 ^ 47531 ^ 1 1 x 4321 ^ 1 1 .
^ ^ 367286 73457 Hx^^a-n 17*
234
ANSWERS TO EXAMIKATION-PAPEB.
Find G.C.M. of terms of second fraction :
47531 )73457 (1
Strike out 2 )25926
12963)47531(3
38889
Strike out 2) 8642
G.C.M.» 4321)1 2963(3
4. (2) L.C.M. of denominators in both terms, 28.
( ^-^^-^T + -ra + !fe)x28 _ 14+ 7 + 4 + 2+l __28 1
Ans.
(J + i + f + i| + |J-)x28 14 + 21 + 24 + 26 + 27"'l]2""4
^ .J. 532 hrs. 357 min. _ 319557 ^ 106519 _ 15217 x 7 7 j
^ ^ 836 hrs. 561 min. 602161 167387 16217 x 11 ~ 11 '
5. (i) 304)12191-92(40105 Ans,
1216
5. (ft)
6.
319
304
1520
1520
Ton
•07890625
20
cwt.
1*57812600
4
qrs.
2-312500
4
1-2600
7
(c)
lbs. 8-75 = 8J lbs.
Ans, 1 cwt. 2 qrs. 8| lbs.
(i) 4112at£l Is. 7i<i.
same as 1028 at £4 6s. 6d,
4
:e4112
bs, = l
ls. = |
rd,=h
257
51
8s. Od,
21
8s. Ad.
Ans. £4441 16s. U,
\
(ii) 121-9192-J-304
=s]st answer -^ 100
= •40105. Ans,
5 of ^ of «?J
3 6J " 41|
_495j^48^5
^260 65^3
_ 56x9x3x 16
"^ 60x55x3
= ^ = 2i|. ^«,.
(ii) Herel9Jyds. =1^
30
or ^^j, or ^ of a pole; ai
we hare to calculate 3 i
1 ro. 3^ po. at £110 p
acre.
£110
3
£330
1 ro.= i
8ipo.=i
27 10s
2 10s.
An&. £%^0 Qs.
ANSWERS TO EXAMINATION-PAPER. 235
7. The leogth, breadth, and height are as 2, 1|, and 1.
No. of square units of wall surface = (2 + 1 J) x 2 x 1 = 7.
No. of square feet of ditto = 105s. -r lid, «'21a. -5- Ji. » 1008.
Hence the square unit= 1008 sq. ft.-5-7 = H4 sq. ft. ; and
the lineal unit = a/144 = 12.
Therefore the required height is 12 fb., length 24 ft., breadth
18 ft. Ans.
8. 5 bojs do the work of 3 men ; or, 40 boys that of 24 men, and 20
boys that of 1 2 men.
Thus the work of 40 boys+ 15 men = that of 39 men,
and the work of 20 boys + 20 men = that of 32 men ;
and the first question is. If 39 men take 8 weeks, how long woiild 32
men take ?
32 m. : 39 m. : : 8 wks. : 9| wks. Ut Ans.
Again: 2 men have the wages of 3 boys; or, 15 men those of 22^
boys, and 20 men those of 30 boys.
Thus 40 boys + 15 men have 62 J times a boy's weekly wages,
and 20 boys + 20 men have 50 times ditto ;
and the second question is, If £350 be the amount of 8 times 62^ pay-
ments, what would be the amount of 9f times 50 such payments ?
62Jx8 : 50x9| :: £350
or 250 X 8 is to 50 X 39
or 40 is to 39.
Hence f§ of £350 = £350 - ^^ of £350,
^ = __8_15s.
£341 5s. 2nd Ans.
9. A is due at his destination in 8 -7-4^ or 1| hours. He walks a
mile at his proper rate in | of an hour ; and ^ an hour is the time
taken by A in walking back a mile at ^'s rate and retracing the mile
at his own proper rate. Hence, the time in which B walks a mile is J
an hour— I of an hour, or /g of an hour; therefore Bb rate per hour is
-^ = 3| miles. 1st Ans.
Again: A when he parted with B had 9 miles to walk in Ij— ^ or
IJ hour, which would have required him to walk 9-f-li or 6 miles an
hour. 2nd Ans.
10. I of 4j = f per cent. Now the interest of the sum is found by
multiplying it by \ and dividing the product by 100 ; and the true dis-
count of the sum is found by multiplying by | and dividing by lOOf.
Thus the interest is -^ of the sum, and the discount is -^ of the sum.
.*. 4^0 -4b-3' or leAoS' o^ the sum = 2^ shillings,
hence the sum = ^ of 1612008. » 403008. » ifl^h. Au^.
285 AXSWEE3 TO EXAMINATION-PAPER.
1 1 . The income in the 6rst instance is ^ of the money he can invest,
and in the second instance ^^ of it.
, 3 3 291-283 3 1 r *i, i.
, . — , or — -t or , or -— — of the money he can
96 97 8x1164 9312 3104 ^
invest is £1 ; and ,'. he has £3104 to invest. Ans,
12. No. of promises to A— ^ the no. of constituents + J of 400 ;
to^=ig „ „ +Jgof400.
No. of votes for ^ = I no. of const. + 200 - 300 = i no. of const. -100;
,. for 5 = 12 „ + 190-100 = Jg „ + 90.
Now the latter of these votes is to exceed the former by 80 ;
that is, ^ the no. of const.'— 100 = 5§ the no. of const + 10 ;
/, ^ the no. of const. = 1 10, or the whole no. = 4400.
-4*svotes= J of 4400-100 = 2100. 1 .
-B's votes =rj§ of 4400+ 90 = 2180. J ' '
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